3.13.65 \(\int \frac {(A+B x) (a+c x^2)^2}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=214 \[ \frac {4 c \sqrt {d+e x} \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6}-\frac {2 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^{3/2}}+\frac {2 \left (a e^2+c d^2\right )^2 (B d-A e)}{5 e^6 (d+e x)^{5/2}}+\frac {4 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6 \sqrt {d+e x}}-\frac {2 c^2 (d+e x)^{3/2} (5 B d-A e)}{3 e^6}+\frac {2 B c^2 (d+e x)^{5/2}}{5 e^6} \]

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Rubi [A]  time = 0.09, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {772} \begin {gather*} \frac {4 c \sqrt {d+e x} \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6}+\frac {4 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6 \sqrt {d+e x}}-\frac {2 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^{3/2}}+\frac {2 \left (a e^2+c d^2\right )^2 (B d-A e)}{5 e^6 (d+e x)^{5/2}}-\frac {2 c^2 (d+e x)^{3/2} (5 B d-A e)}{3 e^6}+\frac {2 B c^2 (d+e x)^{5/2}}{5 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(B*d - A*e)*(c*d^2 + a*e^2)^2)/(5*e^6*(d + e*x)^(5/2)) - (2*(c*d^2 + a*e^2)*(5*B*c*d^2 - 4*A*c*d*e + a*B*e^
2))/(3*e^6*(d + e*x)^(3/2)) + (4*c*(5*B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(e^6*Sqrt[d + e*x]) + (4
*c*(5*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*Sqrt[d + e*x])/e^6 - (2*c^2*(5*B*d - A*e)*(d + e*x)^(3/2))/(3*e^6) + (2*B
*c^2*(d + e*x)^(5/2))/(5*e^6)

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{7/2}} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )^2}{e^5 (d+e x)^{7/2}}+\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{e^5 (d+e x)^{5/2}}+\frac {2 c \left (-5 B c d^3+3 A c d^2 e-3 a B d e^2+a A e^3\right )}{e^5 (d+e x)^{3/2}}-\frac {2 c \left (-5 B c d^2+2 A c d e-a B e^2\right )}{e^5 \sqrt {d+e x}}+\frac {c^2 (-5 B d+A e) \sqrt {d+e x}}{e^5}+\frac {B c^2 (d+e x)^{3/2}}{e^5}\right ) \, dx\\ &=\frac {2 (B d-A e) \left (c d^2+a e^2\right )^2}{5 e^6 (d+e x)^{5/2}}-\frac {2 \left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{3 e^6 (d+e x)^{3/2}}+\frac {4 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right )}{e^6 \sqrt {d+e x}}+\frac {4 c \left (5 B c d^2-2 A c d e+a B e^2\right ) \sqrt {d+e x}}{e^6}-\frac {2 c^2 (5 B d-A e) (d+e x)^{3/2}}{3 e^6}+\frac {2 B c^2 (d+e x)^{5/2}}{5 e^6}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 212, normalized size = 0.99 \begin {gather*} -\frac {2 \left (3 a^2 A e^5+a^2 B e^4 (2 d+5 e x)+2 a A c e^3 \left (8 d^2+20 d e x+15 e^2 x^2\right )-6 a B c e^2 \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )+A c^2 e \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )-B c^2 \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )\right )}{15 e^6 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(3*a^2*A*e^5 + a^2*B*e^4*(2*d + 5*e*x) + 2*a*A*c*e^3*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - 6*a*B*c*e^2*(16*d^3
+ 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) + A*c^2*e*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 40*d*e^3*x^3 -
5*e^4*x^4) - B*c^2*(256*d^5 + 640*d^4*e*x + 480*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 10*d*e^4*x^4 + 3*e^5*x^5)))/(15
*e^6*(d + e*x)^(5/2))

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IntegrateAlgebraic [A]  time = 0.14, size = 301, normalized size = 1.41 \begin {gather*} \frac {2 \left (-3 a^2 A e^5-5 a^2 B e^4 (d+e x)+3 a^2 B d e^4-6 a A c d^2 e^3+20 a A c d e^3 (d+e x)-30 a A c e^3 (d+e x)^2+6 a B c d^3 e^2-30 a B c d^2 e^2 (d+e x)+90 a B c d e^2 (d+e x)^2+30 a B c e^2 (d+e x)^3-3 A c^2 d^4 e+20 A c^2 d^3 e (d+e x)-90 A c^2 d^2 e (d+e x)^2-60 A c^2 d e (d+e x)^3+5 A c^2 e (d+e x)^4+3 B c^2 d^5-25 B c^2 d^4 (d+e x)+150 B c^2 d^3 (d+e x)^2+150 B c^2 d^2 (d+e x)^3-25 B c^2 d (d+e x)^4+3 B c^2 (d+e x)^5\right )}{15 e^6 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(3*B*c^2*d^5 - 3*A*c^2*d^4*e + 6*a*B*c*d^3*e^2 - 6*a*A*c*d^2*e^3 + 3*a^2*B*d*e^4 - 3*a^2*A*e^5 - 25*B*c^2*d
^4*(d + e*x) + 20*A*c^2*d^3*e*(d + e*x) - 30*a*B*c*d^2*e^2*(d + e*x) + 20*a*A*c*d*e^3*(d + e*x) - 5*a^2*B*e^4*
(d + e*x) + 150*B*c^2*d^3*(d + e*x)^2 - 90*A*c^2*d^2*e*(d + e*x)^2 + 90*a*B*c*d*e^2*(d + e*x)^2 - 30*a*A*c*e^3
*(d + e*x)^2 + 150*B*c^2*d^2*(d + e*x)^3 - 60*A*c^2*d*e*(d + e*x)^3 + 30*a*B*c*e^2*(d + e*x)^3 - 25*B*c^2*d*(d
+ e*x)^4 + 5*A*c^2*e*(d + e*x)^4 + 3*B*c^2*(d + e*x)^5))/(15*e^6*(d + e*x)^(5/2))

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fricas [A]  time = 0.42, size = 280, normalized size = 1.31 \begin {gather*} \frac {2 \, {\left (3 \, B c^{2} e^{5} x^{5} + 256 \, B c^{2} d^{5} - 128 \, A c^{2} d^{4} e + 96 \, B a c d^{3} e^{2} - 16 \, A a c d^{2} e^{3} - 2 \, B a^{2} d e^{4} - 3 \, A a^{2} e^{5} - 5 \, {\left (2 \, B c^{2} d e^{4} - A c^{2} e^{5}\right )} x^{4} + 10 \, {\left (8 \, B c^{2} d^{2} e^{3} - 4 \, A c^{2} d e^{4} + 3 \, B a c e^{5}\right )} x^{3} + 30 \, {\left (16 \, B c^{2} d^{3} e^{2} - 8 \, A c^{2} d^{2} e^{3} + 6 \, B a c d e^{4} - A a c e^{5}\right )} x^{2} + 5 \, {\left (128 \, B c^{2} d^{4} e - 64 \, A c^{2} d^{3} e^{2} + 48 \, B a c d^{2} e^{3} - 8 \, A a c d e^{4} - B a^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*B*c^2*e^5*x^5 + 256*B*c^2*d^5 - 128*A*c^2*d^4*e + 96*B*a*c*d^3*e^2 - 16*A*a*c*d^2*e^3 - 2*B*a^2*d*e^4
- 3*A*a^2*e^5 - 5*(2*B*c^2*d*e^4 - A*c^2*e^5)*x^4 + 10*(8*B*c^2*d^2*e^3 - 4*A*c^2*d*e^4 + 3*B*a*c*e^5)*x^3 + 3
0*(16*B*c^2*d^3*e^2 - 8*A*c^2*d^2*e^3 + 6*B*a*c*d*e^4 - A*a*c*e^5)*x^2 + 5*(128*B*c^2*d^4*e - 64*A*c^2*d^3*e^2
+ 48*B*a*c*d^2*e^3 - 8*A*a*c*d*e^4 - B*a^2*e^5)*x)*sqrt(e*x + d)/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e
^6)

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giac [A]  time = 0.21, size = 319, normalized size = 1.49 \begin {gather*} \frac {2}{15} \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B c^{2} e^{24} - 25 \, {\left (x e + d\right )}^{\frac {3}{2}} B c^{2} d e^{24} + 150 \, \sqrt {x e + d} B c^{2} d^{2} e^{24} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{2} e^{25} - 60 \, \sqrt {x e + d} A c^{2} d e^{25} + 30 \, \sqrt {x e + d} B a c e^{26}\right )} e^{\left (-30\right )} + \frac {2 \, {\left (150 \, {\left (x e + d\right )}^{2} B c^{2} d^{3} - 25 \, {\left (x e + d\right )} B c^{2} d^{4} + 3 \, B c^{2} d^{5} - 90 \, {\left (x e + d\right )}^{2} A c^{2} d^{2} e + 20 \, {\left (x e + d\right )} A c^{2} d^{3} e - 3 \, A c^{2} d^{4} e + 90 \, {\left (x e + d\right )}^{2} B a c d e^{2} - 30 \, {\left (x e + d\right )} B a c d^{2} e^{2} + 6 \, B a c d^{3} e^{2} - 30 \, {\left (x e + d\right )}^{2} A a c e^{3} + 20 \, {\left (x e + d\right )} A a c d e^{3} - 6 \, A a c d^{2} e^{3} - 5 \, {\left (x e + d\right )} B a^{2} e^{4} + 3 \, B a^{2} d e^{4} - 3 \, A a^{2} e^{5}\right )} e^{\left (-6\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*(x*e + d)^(5/2)*B*c^2*e^24 - 25*(x*e + d)^(3/2)*B*c^2*d*e^24 + 150*sqrt(x*e + d)*B*c^2*d^2*e^24 + 5*(x
*e + d)^(3/2)*A*c^2*e^25 - 60*sqrt(x*e + d)*A*c^2*d*e^25 + 30*sqrt(x*e + d)*B*a*c*e^26)*e^(-30) + 2/15*(150*(x
*e + d)^2*B*c^2*d^3 - 25*(x*e + d)*B*c^2*d^4 + 3*B*c^2*d^5 - 90*(x*e + d)^2*A*c^2*d^2*e + 20*(x*e + d)*A*c^2*d
^3*e - 3*A*c^2*d^4*e + 90*(x*e + d)^2*B*a*c*d*e^2 - 30*(x*e + d)*B*a*c*d^2*e^2 + 6*B*a*c*d^3*e^2 - 30*(x*e + d
)^2*A*a*c*e^3 + 20*(x*e + d)*A*a*c*d*e^3 - 6*A*a*c*d^2*e^3 - 5*(x*e + d)*B*a^2*e^4 + 3*B*a^2*d*e^4 - 3*A*a^2*e
^5)*e^(-6)/(x*e + d)^(5/2)

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maple [A]  time = 0.05, size = 259, normalized size = 1.21 \begin {gather*} -\frac {2 \left (-3 B \,c^{2} x^{5} e^{5}-5 A \,c^{2} e^{5} x^{4}+10 B \,c^{2} d \,e^{4} x^{4}+40 A \,c^{2} d \,e^{4} x^{3}-30 B a c \,e^{5} x^{3}-80 B \,c^{2} d^{2} e^{3} x^{3}+30 A a c \,e^{5} x^{2}+240 A \,c^{2} d^{2} e^{3} x^{2}-180 B a c d \,e^{4} x^{2}-480 B \,c^{2} d^{3} e^{2} x^{2}+40 A a c d \,e^{4} x +320 A \,c^{2} d^{3} e^{2} x +5 B \,a^{2} e^{5} x -240 B a c \,d^{2} e^{3} x -640 B \,c^{2} d^{4} e x +3 A \,a^{2} e^{5}+16 A \,d^{2} a c \,e^{3}+128 A \,c^{2} d^{4} e +2 B \,a^{2} d \,e^{4}-96 B \,d^{3} a c \,e^{2}-256 B \,c^{2} d^{5}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15/(e*x+d)^(5/2)*(-3*B*c^2*e^5*x^5-5*A*c^2*e^5*x^4+10*B*c^2*d*e^4*x^4+40*A*c^2*d*e^4*x^3-30*B*a*c*e^5*x^3-8
0*B*c^2*d^2*e^3*x^3+30*A*a*c*e^5*x^2+240*A*c^2*d^2*e^3*x^2-180*B*a*c*d*e^4*x^2-480*B*c^2*d^3*e^2*x^2+40*A*a*c*
d*e^4*x+320*A*c^2*d^3*e^2*x+5*B*a^2*e^5*x-240*B*a*c*d^2*e^3*x-640*B*c^2*d^4*e*x+3*A*a^2*e^5+16*A*a*c*d^2*e^3+1
28*A*c^2*d^4*e+2*B*a^2*d*e^4-96*B*a*c*d^3*e^2-256*B*c^2*d^5)/e^6

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maxima [A]  time = 0.57, size = 255, normalized size = 1.19 \begin {gather*} \frac {2 \, {\left (\frac {3 \, {\left (e x + d\right )}^{\frac {5}{2}} B c^{2} - 5 \, {\left (5 \, B c^{2} d - A c^{2} e\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 30 \, {\left (5 \, B c^{2} d^{2} - 2 \, A c^{2} d e + B a c e^{2}\right )} \sqrt {e x + d}}{e^{5}} + \frac {3 \, B c^{2} d^{5} - 3 \, A c^{2} d^{4} e + 6 \, B a c d^{3} e^{2} - 6 \, A a c d^{2} e^{3} + 3 \, B a^{2} d e^{4} - 3 \, A a^{2} e^{5} + 30 \, {\left (5 \, B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 3 \, B a c d e^{2} - A a c e^{3}\right )} {\left (e x + d\right )}^{2} - 5 \, {\left (5 \, B c^{2} d^{4} - 4 \, A c^{2} d^{3} e + 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} + B a^{2} e^{4}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{5}}\right )}}{15 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*((3*(e*x + d)^(5/2)*B*c^2 - 5*(5*B*c^2*d - A*c^2*e)*(e*x + d)^(3/2) + 30*(5*B*c^2*d^2 - 2*A*c^2*d*e + B*a
*c*e^2)*sqrt(e*x + d))/e^5 + (3*B*c^2*d^5 - 3*A*c^2*d^4*e + 6*B*a*c*d^3*e^2 - 6*A*a*c*d^2*e^3 + 3*B*a^2*d*e^4
- 3*A*a^2*e^5 + 30*(5*B*c^2*d^3 - 3*A*c^2*d^2*e + 3*B*a*c*d*e^2 - A*a*c*e^3)*(e*x + d)^2 - 5*(5*B*c^2*d^4 - 4*
A*c^2*d^3*e + 6*B*a*c*d^2*e^2 - 4*A*a*c*d*e^3 + B*a^2*e^4)*(e*x + d))/((e*x + d)^(5/2)*e^5))/e

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mupad [B]  time = 1.79, size = 251, normalized size = 1.17 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e+4\,B\,a\,c\,e^2\right )}{e^6}-\frac {\left (d+e\,x\right )\,\left (\frac {2\,B\,a^2\,e^4}{3}+4\,B\,a\,c\,d^2\,e^2-\frac {8\,A\,a\,c\,d\,e^3}{3}+\frac {10\,B\,c^2\,d^4}{3}-\frac {8\,A\,c^2\,d^3\,e}{3}\right )-{\left (d+e\,x\right )}^2\,\left (20\,B\,c^2\,d^3-12\,A\,c^2\,d^2\,e+12\,B\,a\,c\,d\,e^2-4\,A\,a\,c\,e^3\right )+\frac {2\,A\,a^2\,e^5}{5}-\frac {2\,B\,c^2\,d^5}{5}-\frac {2\,B\,a^2\,d\,e^4}{5}+\frac {2\,A\,c^2\,d^4\,e}{5}+\frac {4\,A\,a\,c\,d^2\,e^3}{5}-\frac {4\,B\,a\,c\,d^3\,e^2}{5}}{e^6\,{\left (d+e\,x\right )}^{5/2}}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^6}+\frac {2\,c^2\,\left (A\,e-5\,B\,d\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^(1/2)*(20*B*c^2*d^2 + 4*B*a*c*e^2 - 8*A*c^2*d*e))/e^6 - ((d + e*x)*((2*B*a^2*e^4)/3 + (10*B*c^2*d^4
)/3 - (8*A*c^2*d^3*e)/3 - (8*A*a*c*d*e^3)/3 + 4*B*a*c*d^2*e^2) - (d + e*x)^2*(20*B*c^2*d^3 - 4*A*a*c*e^3 - 12*
A*c^2*d^2*e + 12*B*a*c*d*e^2) + (2*A*a^2*e^5)/5 - (2*B*c^2*d^5)/5 - (2*B*a^2*d*e^4)/5 + (2*A*c^2*d^4*e)/5 + (4
*A*a*c*d^2*e^3)/5 - (4*B*a*c*d^3*e^2)/5)/(e^6*(d + e*x)^(5/2)) + (2*B*c^2*(d + e*x)^(5/2))/(5*e^6) + (2*c^2*(A
*e - 5*B*d)*(d + e*x)^(3/2))/(3*e^6)

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sympy [A]  time = 4.40, size = 1426, normalized size = 6.66 \begin {gather*} \begin {cases} - \frac {6 A a^{2} e^{5}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {32 A a c d^{2} e^{3}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {80 A a c d e^{4} x}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {60 A a c e^{5} x^{2}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {256 A c^{2} d^{4} e}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {640 A c^{2} d^{3} e^{2} x}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {480 A c^{2} d^{2} e^{3} x^{2}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {80 A c^{2} d e^{4} x^{3}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {10 A c^{2} e^{5} x^{4}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {4 B a^{2} d e^{4}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {10 B a^{2} e^{5} x}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {192 B a c d^{3} e^{2}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {480 B a c d^{2} e^{3} x}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {360 B a c d e^{4} x^{2}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {60 B a c e^{5} x^{3}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {512 B c^{2} d^{5}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {1280 B c^{2} d^{4} e x}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {960 B c^{2} d^{3} e^{2} x^{2}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {160 B c^{2} d^{2} e^{3} x^{3}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {20 B c^{2} d e^{4} x^{4}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {6 B c^{2} e^{5} x^{5}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {A a^{2} x + \frac {2 A a c x^{3}}{3} + \frac {A c^{2} x^{5}}{5} + \frac {B a^{2} x^{2}}{2} + \frac {B a c x^{4}}{2} + \frac {B c^{2} x^{6}}{6}}{d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-6*A*a**2*e**5/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)
) - 32*A*a*c*d**2*e**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) -
80*A*a*c*d*e**4*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 60*
A*a*c*e**5*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 256*A*
c**2*d**4*e/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 640*A*c**2
*d**3*e**2*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 480*A*c**
2*d**2*e**3*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 80*A*
c**2*d*e**4*x**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 10*A*
c**2*e**5*x**4/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 4*B*a**
2*d*e**4/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 10*B*a**2*e**
5*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 192*B*a*c*d**3*e**
2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 480*B*a*c*d**2*e**3*
x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 360*B*a*c*d*e**4*x**
2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 60*B*a*c*e**5*x**3/(
15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 512*B*c**2*d**5/(15*d**
2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 1280*B*c**2*d**4*e*x/(15*d**2
*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 960*B*c**2*d**3*e**2*x**2/(15*
d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 160*B*c**2*d**2*e**3*x**3/
(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 20*B*c**2*d*e**4*x**4/
(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 6*B*c**2*e**5*x**5/(15
*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)), Ne(e, 0)), ((A*a**2*x + 2*
A*a*c*x**3/3 + A*c**2*x**5/5 + B*a**2*x**2/2 + B*a*c*x**4/2 + B*c**2*x**6/6)/d**(7/2), True))

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