\(\int \frac {(A+B x) (a-c x^2)^{3/2}}{(d+e x)^{7/2}} \, dx\) [271]

Optimal result

Integrand size = 27, antiderivative size = 520 \[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=-\frac {8 c \left (32 B c d^3-12 A c d^2 e-29 a B d e^2+9 a A e^3\right ) \sqrt {a-c x^2}}{15 e^4 \left (c d^2-a e^2\right ) \sqrt {d+e x}}-\frac {4 \left (5 a B e^2-4 c d (8 B d-3 A e)-3 c e (8 B d-3 A e) x\right ) \sqrt {a-c x^2}}{15 e^4 (d+e x)^{3/2}}+\frac {2 (8 B d-3 A e+5 B e x) \left (a-c x^2\right )^{3/2}}{15 e^2 (d+e x)^{5/2}}+\frac {8 \sqrt {a} c^{3/2} \left (32 B c d^3-12 A c d^2 e-29 a B d e^2+9 a A e^3\right ) \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{15 e^5 \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}-\frac {8 \sqrt {a} \sqrt {c} \left (32 B c d^2-12 A c d e-5 a B e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{15 e^5 \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:

-8/15*c*(9*A*a*e^3-12*A*c*d^2*e-29*B*a*d*e^2+32*B*c*d^3)*(-c*x^2+a)^(1/2)/ 
e^4/(-a*e^2+c*d^2)/(e*x+d)^(1/2)-4/15*(5*B*a*e^2-4*c*d*(-3*A*e+8*B*d)-3*c* 
e*(-3*A*e+8*B*d)*x)*(-c*x^2+a)^(1/2)/e^4/(e*x+d)^(3/2)+2/15*(5*B*e*x-3*A*e 
+8*B*d)*(-c*x^2+a)^(3/2)/e^2/(e*x+d)^(5/2)+8/15*a^(1/2)*c^(3/2)*(9*A*a*e^3 
-12*A*c*d^2*e-29*B*a*d*e^2+32*B*c*d^3)*(e*x+d)^(1/2)*(1-c*x^2/a)^(1/2)*Ell 
ipticE(1/2*(1-c^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*e/(c^(1/2) 
*d+a^(1/2)*e))^(1/2))/e^5/(-a*e^2+c*d^2)/(c^(1/2)*(e*x+d)/(c^(1/2)*d+a^(1/ 
2)*e))^(1/2)/(-c*x^2+a)^(1/2)-8/15*a^(1/2)*c^(1/2)*(-12*A*c*d*e-5*B*a*e^2+ 
32*B*c*d^2)*(c^(1/2)*(e*x+d)/(c^(1/2)*d+a^(1/2)*e))^(1/2)*(1-c*x^2/a)^(1/2 
)*EllipticF(1/2*(1-c^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*e/(c^ 
(1/2)*d+a^(1/2)*e))^(1/2))/e^5/(e*x+d)^(1/2)/(-c*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 26.86 (sec) , antiderivative size = 672, normalized size of antiderivative = 1.29 \[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\frac {\sqrt {a-c x^2} \left (\frac {2 (d+e x) \left (-5 B c-\frac {3 (B d-A e) \left (c d^2-a e^2\right )}{(d+e x)^3}+\frac {17 B c d^2-12 A c d e-5 a B e^2}{(d+e x)^2}+\frac {c \left (73 B c d^3-33 A c d^2 e-61 a B d e^2+21 a A e^3\right )}{\left (-c d^2+a e^2\right ) (d+e x)}\right )}{e^4}+\frac {8 c \left (e^2 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (32 B c d^3-12 A c d^2 e-29 a B d e^2+9 a A e^3\right ) \left (a-c x^2\right )+i \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (32 B c d^3-12 A c d^2 e-29 a B d e^2+9 a A e^3\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )+i \sqrt {a} e \left (-\sqrt {c} d+\sqrt {a} e\right ) \left (-32 B c d^2-24 \sqrt {a} B \sqrt {c} d e+12 A c d e+5 a B e^2+9 \sqrt {a} A \sqrt {c} e^2\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )\right )}{e^6 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-c d^2+a e^2\right ) \left (-a+c x^2\right )}\right )}{15 \sqrt {d+e x}} \] Input:

Integrate[((A + B*x)*(a - c*x^2)^(3/2))/(d + e*x)^(7/2),x]
 

Output:

(Sqrt[a - c*x^2]*((2*(d + e*x)*(-5*B*c - (3*(B*d - A*e)*(c*d^2 - a*e^2))/( 
d + e*x)^3 + (17*B*c*d^2 - 12*A*c*d*e - 5*a*B*e^2)/(d + e*x)^2 + (c*(73*B* 
c*d^3 - 33*A*c*d^2*e - 61*a*B*d*e^2 + 21*a*A*e^3))/((-(c*d^2) + a*e^2)*(d 
+ e*x))))/e^4 + (8*c*(e^2*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(32*B*c*d^3 - 12* 
A*c*d^2*e - 29*a*B*d*e^2 + 9*a*A*e^3)*(a - c*x^2) + I*Sqrt[c]*(Sqrt[c]*d - 
 Sqrt[a]*e)*(32*B*c*d^3 - 12*A*c*d^2*e - 29*a*B*d*e^2 + 9*a*A*e^3)*Sqrt[(e 
*(Sqrt[a]/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((Sqrt[a]*e)/Sqrt[c] - e*x)/(d + 
 e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]] 
/Sqrt[d + e*x]], (Sqrt[c]*d + Sqrt[a]*e)/(Sqrt[c]*d - Sqrt[a]*e)] + I*Sqrt 
[a]*e*(-(Sqrt[c]*d) + Sqrt[a]*e)*(-32*B*c*d^2 - 24*Sqrt[a]*B*Sqrt[c]*d*e + 
 12*A*c*d*e + 5*a*B*e^2 + 9*Sqrt[a]*A*Sqrt[c]*e^2)*Sqrt[(e*(Sqrt[a]/Sqrt[c 
] + x))/(d + e*x)]*Sqrt[-(((Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x 
)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], 
 (Sqrt[c]*d + Sqrt[a]*e)/(Sqrt[c]*d - Sqrt[a]*e)]))/(e^6*Sqrt[-d + (Sqrt[a 
]*e)/Sqrt[c]]*(-(c*d^2) + a*e^2)*(-a + c*x^2))))/(15*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {680, 25, 27, 681, 25, 600, 509, 508, 327, 512, 511, 321}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a-c x^2\right )^{3/2} (A+B x)}{(d+e x)^{7/2}} \, dx\)

\(\Big \downarrow \) 680

\(\displaystyle \frac {2 \int -\frac {c \left (3 a e (B d-A e)+\left (8 B c d^2-3 A c e d-5 a B e^2\right ) x\right ) \sqrt {a-c x^2}}{(d+e x)^{3/2}}dx}{5 e^2 \left (c d^2-a e^2\right )}+\frac {2 \left (a-c x^2\right )^{3/2} \left (-e x \left (-5 a B e^2-6 A c d e+11 B c d^2\right )+3 A e \left (a e^2+c d^2\right )-2 B \left (4 c d^3-a d e^2\right )\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \left (a-c x^2\right )^{3/2} \left (-e x \left (-5 a B e^2-6 A c d e+11 B c d^2\right )+3 A e \left (a e^2+c d^2\right )-2 B \left (4 c d^3-a d e^2\right )\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {2 \int \frac {c \left (3 a e (B d-A e)+\left (8 B c d^2-3 A c e d-5 a B e^2\right ) x\right ) \sqrt {a-c x^2}}{(d+e x)^{3/2}}dx}{5 e^2 \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (a-c x^2\right )^{3/2} \left (-e x \left (-5 a B e^2-6 A c d e+11 B c d^2\right )+3 A e \left (a e^2+c d^2\right )-2 B \left (4 c d^3-a d e^2\right )\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {2 c \int \frac {\left (3 a e (B d-A e)+\left (8 B c d^2-3 A c e d-5 a B e^2\right ) x\right ) \sqrt {a-c x^2}}{(d+e x)^{3/2}}dx}{5 e^2 \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 681

\(\displaystyle \frac {2 \left (a-c x^2\right )^{3/2} \left (-e x \left (-5 a B e^2-6 A c d e+11 B c d^2\right )+3 A e \left (a e^2+c d^2\right )-2 B \left (4 c d^3-a d e^2\right )\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {2 c \left (\frac {2 \sqrt {a-c x^2} \left (e x \left (-5 a B e^2-3 A c d e+8 B c d^2\right )+9 a A e^3-29 a B d e^2-12 A c d^2 e+32 B c d^3\right )}{3 e^2 \sqrt {d+e x}}-\frac {2 \int -\frac {a e \left (8 B c d^2-3 A c e d-5 a B e^2\right )+c \left (32 B c d^3-12 A c e d^2-29 a B e^2 d+9 a A e^3\right ) x}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{3 e^2}\right )}{5 e^2 \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \left (a-c x^2\right )^{3/2} \left (-e x \left (-5 a B e^2-6 A c d e+11 B c d^2\right )+3 A e \left (a e^2+c d^2\right )-2 B \left (4 c d^3-a d e^2\right )\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {2 c \left (\frac {2 \int \frac {a e \left (8 B c d^2-3 A c e d-5 a B e^2\right )+c \left (32 B c d^3-12 A c e d^2-29 a B e^2 d+9 a A e^3\right ) x}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{3 e^2}+\frac {2 \sqrt {a-c x^2} \left (e x \left (-5 a B e^2-3 A c d e+8 B c d^2\right )+9 a A e^3-29 a B d e^2-12 A c d^2 e+32 B c d^3\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2 \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {2 \left (a-c x^2\right )^{3/2} \left (-e x \left (-5 a B e^2-6 A c d e+11 B c d^2\right )+3 A e \left (a e^2+c d^2\right )-2 B \left (4 c d^3-a d e^2\right )\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {2 c \left (\frac {2 \left (\frac {c \left (9 a A e^3-29 a B d e^2-12 A c d^2 e+32 B c d^3\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a-c x^2}}dx}{e}-\frac {\left (c d^2-a e^2\right ) \left (-5 a B e^2-12 A c d e+32 B c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{3 e^2}+\frac {2 \sqrt {a-c x^2} \left (e x \left (-5 a B e^2-3 A c d e+8 B c d^2\right )+9 a A e^3-29 a B d e^2-12 A c d^2 e+32 B c d^3\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2 \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {2 \left (a-c x^2\right )^{3/2} \left (-e x \left (-5 a B e^2-6 A c d e+11 B c d^2\right )+3 A e \left (a e^2+c d^2\right )-2 B \left (4 c d^3-a d e^2\right )\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {2 c \left (\frac {2 \left (\frac {c \sqrt {1-\frac {c x^2}{a}} \left (9 a A e^3-29 a B d e^2-12 A c d^2 e+32 B c d^3\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {\left (c d^2-a e^2\right ) \left (-5 a B e^2-12 A c d e+32 B c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{3 e^2}+\frac {2 \sqrt {a-c x^2} \left (e x \left (-5 a B e^2-3 A c d e+8 B c d^2\right )+9 a A e^3-29 a B d e^2-12 A c d^2 e+32 B c d^3\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2 \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {2 \left (a-c x^2\right )^{3/2} \left (-e x \left (-5 a B e^2-6 A c d e+11 B c d^2\right )+3 A e \left (a e^2+c d^2\right )-2 B \left (4 c d^3-a d e^2\right )\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {2 c \left (\frac {2 \left (-\frac {\left (c d^2-a e^2\right ) \left (-5 a B e^2-12 A c d e+32 B c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (9 a A e^3-29 a B d e^2-12 A c d^2 e+32 B c d^3\right ) \int \frac {\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{3 e^2}+\frac {2 \sqrt {a-c x^2} \left (e x \left (-5 a B e^2-3 A c d e+8 B c d^2\right )+9 a A e^3-29 a B d e^2-12 A c d^2 e+32 B c d^3\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2 \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2 \left (a-c x^2\right )^{3/2} \left (-e x \left (-5 a B e^2-6 A c d e+11 B c d^2\right )+3 A e \left (a e^2+c d^2\right )-2 B \left (4 c d^3-a d e^2\right )\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {2 c \left (\frac {2 \left (-\frac {\left (c d^2-a e^2\right ) \left (-5 a B e^2-12 A c d e+32 B c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (9 a A e^3-29 a B d e^2-12 A c d^2 e+32 B c d^3\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{3 e^2}+\frac {2 \sqrt {a-c x^2} \left (e x \left (-5 a B e^2-3 A c d e+8 B c d^2\right )+9 a A e^3-29 a B d e^2-12 A c d^2 e+32 B c d^3\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2 \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {2 \left (a-c x^2\right )^{3/2} \left (-e x \left (-5 a B e^2-6 A c d e+11 B c d^2\right )+3 A e \left (a e^2+c d^2\right )-2 B \left (4 c d^3-a d e^2\right )\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {2 c \left (\frac {2 \left (-\frac {\sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \left (-5 a B e^2-12 A c d e+32 B c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (9 a A e^3-29 a B d e^2-12 A c d^2 e+32 B c d^3\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{3 e^2}+\frac {2 \sqrt {a-c x^2} \left (e x \left (-5 a B e^2-3 A c d e+8 B c d^2\right )+9 a A e^3-29 a B d e^2-12 A c d^2 e+32 B c d^3\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2 \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {2 \left (a-c x^2\right )^{3/2} \left (-e x \left (-5 a B e^2-6 A c d e+11 B c d^2\right )+3 A e \left (a e^2+c d^2\right )-2 B \left (4 c d^3-a d e^2\right )\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {2 c \left (\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \left (-5 a B e^2-12 A c d e+32 B c d^2\right ) \int \frac {1}{\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (9 a A e^3-29 a B d e^2-12 A c d^2 e+32 B c d^3\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{3 e^2}+\frac {2 \sqrt {a-c x^2} \left (e x \left (-5 a B e^2-3 A c d e+8 B c d^2\right )+9 a A e^3-29 a B d e^2-12 A c d^2 e+32 B c d^3\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2 \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2 \left (a-c x^2\right )^{3/2} \left (-e x \left (-5 a B e^2-6 A c d e+11 B c d^2\right )+3 A e \left (a e^2+c d^2\right )-2 B \left (4 c d^3-a d e^2\right )\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {2 c \left (\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \left (-5 a B e^2-12 A c d e+32 B c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (9 a A e^3-29 a B d e^2-12 A c d^2 e+32 B c d^3\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{3 e^2}+\frac {2 \sqrt {a-c x^2} \left (e x \left (-5 a B e^2-3 A c d e+8 B c d^2\right )+9 a A e^3-29 a B d e^2-12 A c d^2 e+32 B c d^3\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2 \left (c d^2-a e^2\right )}\)

Input:

Int[((A + B*x)*(a - c*x^2)^(3/2))/(d + e*x)^(7/2),x]
 

Output:

(2*(3*A*e*(c*d^2 + a*e^2) - 2*B*(4*c*d^3 - a*d*e^2) - e*(11*B*c*d^2 - 6*A* 
c*d*e - 5*a*B*e^2)*x)*(a - c*x^2)^(3/2))/(15*e^2*(c*d^2 - a*e^2)*(d + e*x) 
^(5/2)) - (2*c*((2*(32*B*c*d^3 - 12*A*c*d^2*e - 29*a*B*d*e^2 + 9*a*A*e^3 + 
 e*(8*B*c*d^2 - 3*A*c*d*e - 5*a*B*e^2)*x)*Sqrt[a - c*x^2])/(3*e^2*Sqrt[d + 
 e*x]) + (2*((-2*Sqrt[a]*Sqrt[c]*(32*B*c*d^3 - 12*A*c*d^2*e - 29*a*B*d*e^2 
 + 9*a*A*e^3)*Sqrt[d + e*x]*Sqrt[1 - (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - 
(Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*e)/((Sqrt[c]*d)/Sqrt[a] + e)])/(e*Sqrt[( 
Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[a - c*x^2]) + (2*Sqrt[a]* 
(c*d^2 - a*e^2)*(32*B*c*d^2 - 12*A*c*d*e - 5*a*B*e^2)*Sqrt[(Sqrt[c]*(d + e 
*x))/(Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[1 - (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 
- (Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*e)/((Sqrt[c]*d)/Sqrt[a] + e)])/(Sqrt[c 
]*e*Sqrt[d + e*x]*Sqrt[a - c*x^2])))/(3*e^2)))/(5*e^2*(c*d^2 - a*e^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 508

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q 
 = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c 
*q))]))   Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr 
t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
 

rule 509

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq 
rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], 
x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 511

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt 
[c + d*x]))   Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] 
, x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 512

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 
2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 600

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[B/d   Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp 
[(B*c - A*d)/d   Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, 
 b, c, d, A, B}, x] && NegQ[b/a]
 

rule 680

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + c*x^2)^p/(e^2*(m + 1)*(m 
+ 2)*(c*d^2 + a*e^2)))*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e* 
f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e^2) + 2*c*d*p*(e*f - d*g))*x), x] - Sim 
p[p/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2))   Int[(d + e*x)^(m + 2)*(a + c*x^ 
2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1) - e*f 
*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, 
 g}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3 
, 0]
 

rule 681

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) 
 + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/ 
(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Sim 
p[g*(2*a*e + 2*a*e*m) + (g*(2*c*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x] 
, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, -1] || 
EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ[m + 2 
*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(972\) vs. \(2(444)=888\).

Time = 17.20 (sec) , antiderivative size = 973, normalized size of antiderivative = 1.87

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}\, \left (-\frac {2 \left (A a \,e^{3}-A c \,d^{2} e -B a d \,e^{2}+B c \,d^{3}\right ) \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}{5 e^{7} \left (x +\frac {d}{e}\right )^{3}}-\frac {2 \left (12 A c d e +5 B a \,e^{2}-17 B c \,d^{2}\right ) \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}{15 e^{6} \left (x +\frac {d}{e}\right )^{2}}+\frac {2 \left (-c e \,x^{2}+a e \right ) c \left (21 A a \,e^{3}-33 A c \,d^{2} e -61 B a d \,e^{2}+73 B c \,d^{3}\right )}{15 e^{5} \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right ) \left (-c e \,x^{2}+a e \right )}}-\frac {2 B c \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}{3 e^{4}}+\frac {2 \left (-\frac {c \left (3 A c d e +2 B a \,e^{2}-6 B c \,d^{2}\right )}{e^{5}}+\frac {c \left (12 A c d e +5 B a \,e^{2}-17 B c \,d^{2}\right )}{15 e^{5}}+\frac {c^{2} d \left (21 A a \,e^{3}-33 A c \,d^{2} e -61 B a d \,e^{2}+73 B c \,d^{3}\right )}{15 e^{5} \left (a \,e^{2}-c \,d^{2}\right )}+\frac {B c a}{3 e^{3}}\right ) \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}+\frac {2 \left (\frac {c^{2} \left (A e -3 B d \right )}{e^{4}}+\frac {c^{2} \left (21 A a \,e^{3}-33 A c \,d^{2} e -61 B a d \,e^{2}+73 B c \,d^{3}\right )}{15 e^{4} \left (a \,e^{2}-c \,d^{2}\right )}-\frac {2 B \,c^{2} d}{3 e^{4}}\right ) \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )+\frac {\sqrt {a c}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{c}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}\)	\(973\)
risch	\(\text {Expression too large to display}\)	\(2346\)
default	\(\text {Expression too large to display}\)	\(7010\)

Input:

int((B*x+A)*(-c*x^2+a)^(3/2)/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

((e*x+d)*(-c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(-c*x^2+a)^(1/2)*(-2/5*(A*a*e^3-A 
*c*d^2*e-B*a*d*e^2+B*c*d^3)/e^7*(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)/(x+d/e) 
^3-2/15*(12*A*c*d*e+5*B*a*e^2-17*B*c*d^2)/e^6*(-c*e*x^3-c*d*x^2+a*e*x+a*d) 
^(1/2)/(x+d/e)^2+2/15*(-c*e*x^2+a*e)/e^5/(a*e^2-c*d^2)*c*(21*A*a*e^3-33*A* 
c*d^2*e-61*B*a*d*e^2+73*B*c*d^3)/((x+d/e)*(-c*e*x^2+a*e))^(1/2)-2/3*B/e^4* 
c*(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)+2*(-c*(3*A*c*d*e+2*B*a*e^2-6*B*c*d^2) 
/e^5+1/15*c*(12*A*c*d*e+5*B*a*e^2-17*B*c*d^2)/e^5+1/15*c^2/e^5*d*(21*A*a*e 
^3-33*A*c*d^2*e-61*B*a*d*e^2+73*B*c*d^3)/(a*e^2-c*d^2)+1/3*B/e^3*c*a)*(d/e 
-1/c*(a*c)^(1/2))*((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2)*((x-1/c*(a*c)^(1/2 
))/(-d/e-1/c*(a*c)^(1/2)))^(1/2)*((x+1/c*(a*c)^(1/2))/(-d/e+1/c*(a*c)^(1/2 
)))^(1/2)/(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/e-1/c*( 
a*c)^(1/2)))^(1/2),((-d/e+1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1/2))+ 
2*(1/e^4*c^2*(A*e-3*B*d)+1/15*c^2/e^4*(21*A*a*e^3-33*A*c*d^2*e-61*B*a*d*e^ 
2+73*B*c*d^3)/(a*e^2-c*d^2)-2/3*B/e^4*c^2*d)*(d/e-1/c*(a*c)^(1/2))*((x+d/e 
)/(d/e-1/c*(a*c)^(1/2)))^(1/2)*((x-1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)) 
)^(1/2)*((x+1/c*(a*c)^(1/2))/(-d/e+1/c*(a*c)^(1/2)))^(1/2)/(-c*e*x^3-c*d*x 
^2+a*e*x+a*d)^(1/2)*((-d/e-1/c*(a*c)^(1/2))*EllipticE(((x+d/e)/(d/e-1/c*(a 
*c)^(1/2)))^(1/2),((-d/e+1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1/2))+1 
/c*(a*c)^(1/2)*EllipticF(((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2),((-d/e+1/c* 
(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 885, normalized size of antiderivative = 1.70 \[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(-c*x^2+a)^(3/2)/(e*x+d)^(7/2),x, algorithm="fricas")
 

Output:

-2/45*(4*(32*B*c^2*d^7 - 12*A*c^2*d^6*e - 53*B*a*c*d^5*e^2 + 18*A*a*c*d^4* 
e^3 + 15*B*a^2*d^3*e^4 + (32*B*c^2*d^4*e^3 - 12*A*c^2*d^3*e^4 - 53*B*a*c*d 
^2*e^5 + 18*A*a*c*d*e^6 + 15*B*a^2*e^7)*x^3 + 3*(32*B*c^2*d^5*e^2 - 12*A*c 
^2*d^4*e^3 - 53*B*a*c*d^3*e^4 + 18*A*a*c*d^2*e^5 + 15*B*a^2*d*e^6)*x^2 + 3 
*(32*B*c^2*d^6*e - 12*A*c^2*d^5*e^2 - 53*B*a*c*d^4*e^3 + 18*A*a*c*d^3*e^4 
+ 15*B*a^2*d^2*e^5)*x)*sqrt(-c*e)*weierstrassPInverse(4/3*(c*d^2 + 3*a*e^2 
)/(c*e^2), -8/27*(c*d^3 - 9*a*d*e^2)/(c*e^3), 1/3*(3*e*x + d)/e) + 12*(32* 
B*c^2*d^6*e - 12*A*c^2*d^5*e^2 - 29*B*a*c*d^4*e^3 + 9*A*a*c*d^3*e^4 + (32* 
B*c^2*d^3*e^4 - 12*A*c^2*d^2*e^5 - 29*B*a*c*d*e^6 + 9*A*a*c*e^7)*x^3 + 3*( 
32*B*c^2*d^4*e^3 - 12*A*c^2*d^3*e^4 - 29*B*a*c*d^2*e^5 + 9*A*a*c*d*e^6)*x^ 
2 + 3*(32*B*c^2*d^5*e^2 - 12*A*c^2*d^4*e^3 - 29*B*a*c*d^3*e^4 + 9*A*a*c*d^ 
2*e^5)*x)*sqrt(-c*e)*weierstrassZeta(4/3*(c*d^2 + 3*a*e^2)/(c*e^2), -8/27* 
(c*d^3 - 9*a*d*e^2)/(c*e^3), weierstrassPInverse(4/3*(c*d^2 + 3*a*e^2)/(c* 
e^2), -8/27*(c*d^3 - 9*a*d*e^2)/(c*e^3), 1/3*(3*e*x + d)/e)) + 3*(64*B*c^2 
*d^5*e^2 - 24*A*c^2*d^4*e^3 - 50*B*a*c*d^3*e^4 + 15*A*a*c*d^2*e^5 - 2*B*a^ 
2*d*e^6 - 3*A*a^2*e^7 + 5*(B*c^2*d^2*e^5 - B*a*c*e^7)*x^3 + (88*B*c^2*d^3* 
e^4 - 33*A*c^2*d^2*e^5 - 76*B*a*c*d*e^6 + 21*A*a*c*e^7)*x^2 + (144*B*c^2*d 
^4*e^3 - 54*A*c^2*d^3*e^4 - 115*B*a*c*d^2*e^5 + 30*A*a*c*d*e^6 - 5*B*a^2*e 
^7)*x)*sqrt(-c*x^2 + a)*sqrt(e*x + d))/(c*d^5*e^6 - a*d^3*e^8 + (c*d^2*e^9 
 - a*e^11)*x^3 + 3*(c*d^3*e^8 - a*d*e^10)*x^2 + 3*(c*d^4*e^7 - a*d^2*e^...
 

Sympy [F]

\[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (A + B x\right ) \left (a - c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((B*x+A)*(-c*x**2+a)**(3/2)/(e*x+d)**(7/2),x)
 

Output:

Integral((A + B*x)*(a - c*x**2)**(3/2)/(d + e*x)**(7/2), x)
 

Maxima [F]

\[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (-c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((B*x+A)*(-c*x^2+a)^(3/2)/(e*x+d)^(7/2),x, algorithm="maxima")
 

Output:

integrate((-c*x^2 + a)^(3/2)*(B*x + A)/(e*x + d)^(7/2), x)
 

Giac [F]

\[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (-c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((B*x+A)*(-c*x^2+a)^(3/2)/(e*x+d)^(7/2),x, algorithm="giac")
 

Output:

integrate((-c*x^2 + a)^(3/2)*(B*x + A)/(e*x + d)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int \frac {{\left (a-c\,x^2\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^{7/2}} \,d x \] Input:

int(((a - c*x^2)^(3/2)*(A + B*x))/(d + e*x)^(7/2),x)
 

Output:

int(((a - c*x^2)^(3/2)*(A + B*x))/(d + e*x)^(7/2), x)
 

Reduce [F]

\[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\text {too large to display} \] Input:

int((B*x+A)*(-c*x^2+a)^(3/2)/(e*x+d)^(7/2),x)
                                                                                    
                                                                                    
 

Output:

(2*(6*sqrt(d + e*x)*sqrt(a - c*x**2)*a**2*b*e**3 + 21*sqrt(d + e*x)*sqrt(a 
 - c*x**2)*a**2*c*d*e**2 - 56*sqrt(d + e*x)*sqrt(a - c*x**2)*a*b*c*d**2*e 
- 5*sqrt(d + e*x)*sqrt(a - c*x**2)*a*b*c*d*e**2*x - 18*sqrt(d + e*x)*sqrt( 
a - c*x**2)*a*c**2*d**2*e*x - 3*sqrt(d + e*x)*sqrt(a - c*x**2)*a*c**2*d*e* 
*2*x**2 + 48*sqrt(d + e*x)*sqrt(a - c*x**2)*b*c**2*d**3*x + 8*sqrt(d + e*x 
)*sqrt(a - c*x**2)*b*c**2*d**2*e*x**2 - sqrt(d + e*x)*sqrt(a - c*x**2)*b*c 
**2*d*e**2*x**3 - 9*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d**4 + 4* 
a*d**3*e*x + 6*a*d**2*e**2*x**2 + 4*a*d*e**3*x**3 + a*e**4*x**4 - c*d**4*x 
**2 - 4*c*d**3*e*x**3 - 6*c*d**2*e**2*x**4 - 4*c*d*e**3*x**5 - c*e**4*x**6 
),x)*a**2*b*c*d**3*e**4 - 27*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a* 
d**4 + 4*a*d**3*e*x + 6*a*d**2*e**2*x**2 + 4*a*d*e**3*x**3 + a*e**4*x**4 - 
 c*d**4*x**2 - 4*c*d**3*e*x**3 - 6*c*d**2*e**2*x**4 - 4*c*d*e**3*x**5 - c* 
e**4*x**6),x)*a**2*b*c*d**2*e**5*x - 27*int((sqrt(d + e*x)*sqrt(a - c*x**2 
)*x**2)/(a*d**4 + 4*a*d**3*e*x + 6*a*d**2*e**2*x**2 + 4*a*d*e**3*x**3 + a* 
e**4*x**4 - c*d**4*x**2 - 4*c*d**3*e*x**3 - 6*c*d**2*e**2*x**4 - 4*c*d*e** 
3*x**5 - c*e**4*x**6),x)*a**2*b*c*d*e**6*x**2 - 9*int((sqrt(d + e*x)*sqrt( 
a - c*x**2)*x**2)/(a*d**4 + 4*a*d**3*e*x + 6*a*d**2*e**2*x**2 + 4*a*d*e**3 
*x**3 + a*e**4*x**4 - c*d**4*x**2 - 4*c*d**3*e*x**3 - 6*c*d**2*e**2*x**4 - 
 4*c*d*e**3*x**5 - c*e**4*x**6),x)*a**2*b*c*e**7*x**3 - 36*int((sqrt(d + e 
*x)*sqrt(a - c*x**2)*x**2)/(a*d**4 + 4*a*d**3*e*x + 6*a*d**2*e**2*x**2 ...