3.6.29 \(\int \frac {(a+c x^2)^3}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=198 \[ \frac {6 c^2 (d+e x)^{7/2} \left (a e^2+5 c d^2\right )}{7 e^7}-\frac {8 c^2 d (d+e x)^{5/2} \left (3 a e^2+5 c d^2\right )}{5 e^7}+\frac {2 c (d+e x)^{3/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7}-\frac {12 c d \sqrt {d+e x} \left (a e^2+c d^2\right )^2}{e^7}-\frac {2 \left (a e^2+c d^2\right )^3}{e^7 \sqrt {d+e x}}+\frac {2 c^3 (d+e x)^{11/2}}{11 e^7}-\frac {4 c^3 d (d+e x)^{9/2}}{3 e^7} \]

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Rubi [A]  time = 0.08, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {697} \begin {gather*} \frac {6 c^2 (d+e x)^{7/2} \left (a e^2+5 c d^2\right )}{7 e^7}-\frac {8 c^2 d (d+e x)^{5/2} \left (3 a e^2+5 c d^2\right )}{5 e^7}+\frac {2 c (d+e x)^{3/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7}-\frac {12 c d \sqrt {d+e x} \left (a e^2+c d^2\right )^2}{e^7}-\frac {2 \left (a e^2+c d^2\right )^3}{e^7 \sqrt {d+e x}}+\frac {2 c^3 (d+e x)^{11/2}}{11 e^7}-\frac {4 c^3 d (d+e x)^{9/2}}{3 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^3/(d + e*x)^(3/2),x]

[Out]

(-2*(c*d^2 + a*e^2)^3)/(e^7*Sqrt[d + e*x]) - (12*c*d*(c*d^2 + a*e^2)^2*Sqrt[d + e*x])/e^7 + (2*c*(c*d^2 + a*e^
2)*(5*c*d^2 + a*e^2)*(d + e*x)^(3/2))/e^7 - (8*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(5/2))/(5*e^7) + (6*c^2*(5*
c*d^2 + a*e^2)*(d + e*x)^(7/2))/(7*e^7) - (4*c^3*d*(d + e*x)^(9/2))/(3*e^7) + (2*c^3*(d + e*x)^(11/2))/(11*e^7
)

Rule 697

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

Rubi steps

\begin {align*} \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{3/2}} \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^3}{e^6 (d+e x)^{3/2}}-\frac {6 c d \left (c d^2+a e^2\right )^2}{e^6 \sqrt {d+e x}}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) \sqrt {d+e x}}{e^6}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{3/2}}{e^6}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{5/2}}{e^6}-\frac {6 c^3 d (d+e x)^{7/2}}{e^6}+\frac {c^3 (d+e x)^{9/2}}{e^6}\right ) \, dx\\ &=-\frac {2 \left (c d^2+a e^2\right )^3}{e^7 \sqrt {d+e x}}-\frac {12 c d \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}{e^7}+\frac {2 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{3/2}}{e^7}-\frac {8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{5/2}}{5 e^7}+\frac {6 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{7/2}}{7 e^7}-\frac {4 c^3 d (d+e x)^{9/2}}{3 e^7}+\frac {2 c^3 (d+e x)^{11/2}}{11 e^7}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 171, normalized size = 0.86 \begin {gather*} -\frac {2 \left (1155 a^3 e^6+1155 a^2 c e^4 \left (8 d^2+4 d e x-e^2 x^2\right )+99 a c^2 e^2 \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )+5 c^3 \left (1024 d^6+512 d^5 e x-128 d^4 e^2 x^2+64 d^3 e^3 x^3-40 d^2 e^4 x^4+28 d e^5 x^5-21 e^6 x^6\right )\right )}{1155 e^7 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1155*a^3*e^6 + 1155*a^2*c*e^4*(8*d^2 + 4*d*e*x - e^2*x^2) + 99*a*c^2*e^2*(128*d^4 + 64*d^3*e*x - 16*d^2*e
^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4) + 5*c^3*(1024*d^6 + 512*d^5*e*x - 128*d^4*e^2*x^2 + 64*d^3*e^3*x^3 - 40*d^2*
e^4*x^4 + 28*d*e^5*x^5 - 21*e^6*x^6)))/(1155*e^7*Sqrt[d + e*x])

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IntegrateAlgebraic [A]  time = 0.08, size = 240, normalized size = 1.21 \begin {gather*} \frac {2 \left (-1155 a^3 e^6-3465 a^2 c d^2 e^4-6930 a^2 c d e^4 (d+e x)+1155 a^2 c e^4 (d+e x)^2-3465 a c^2 d^4 e^2-13860 a c^2 d^3 e^2 (d+e x)+6930 a c^2 d^2 e^2 (d+e x)^2-2772 a c^2 d e^2 (d+e x)^3+495 a c^2 e^2 (d+e x)^4-1155 c^3 d^6-6930 c^3 d^5 (d+e x)+5775 c^3 d^4 (d+e x)^2-4620 c^3 d^3 (d+e x)^3+2475 c^3 d^2 (d+e x)^4-770 c^3 d (d+e x)^5+105 c^3 (d+e x)^6\right )}{1155 e^7 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + c*x^2)^3/(d + e*x)^(3/2),x]

[Out]

(2*(-1155*c^3*d^6 - 3465*a*c^2*d^4*e^2 - 3465*a^2*c*d^2*e^4 - 1155*a^3*e^6 - 6930*c^3*d^5*(d + e*x) - 13860*a*
c^2*d^3*e^2*(d + e*x) - 6930*a^2*c*d*e^4*(d + e*x) + 5775*c^3*d^4*(d + e*x)^2 + 6930*a*c^2*d^2*e^2*(d + e*x)^2
+ 1155*a^2*c*e^4*(d + e*x)^2 - 4620*c^3*d^3*(d + e*x)^3 - 2772*a*c^2*d*e^2*(d + e*x)^3 + 2475*c^3*d^2*(d + e*
x)^4 + 495*a*c^2*e^2*(d + e*x)^4 - 770*c^3*d*(d + e*x)^5 + 105*c^3*(d + e*x)^6))/(1155*e^7*Sqrt[d + e*x])

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fricas [A]  time = 0.39, size = 211, normalized size = 1.07 \begin {gather*} \frac {2 \, {\left (105 \, c^{3} e^{6} x^{6} - 140 \, c^{3} d e^{5} x^{5} - 5120 \, c^{3} d^{6} - 12672 \, a c^{2} d^{4} e^{2} - 9240 \, a^{2} c d^{2} e^{4} - 1155 \, a^{3} e^{6} + 5 \, {\left (40 \, c^{3} d^{2} e^{4} + 99 \, a c^{2} e^{6}\right )} x^{4} - 8 \, {\left (40 \, c^{3} d^{3} e^{3} + 99 \, a c^{2} d e^{5}\right )} x^{3} + {\left (640 \, c^{3} d^{4} e^{2} + 1584 \, a c^{2} d^{2} e^{4} + 1155 \, a^{2} c e^{6}\right )} x^{2} - 4 \, {\left (640 \, c^{3} d^{5} e + 1584 \, a c^{2} d^{3} e^{3} + 1155 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}}{1155 \, {\left (e^{8} x + d e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*(105*c^3*e^6*x^6 - 140*c^3*d*e^5*x^5 - 5120*c^3*d^6 - 12672*a*c^2*d^4*e^2 - 9240*a^2*c*d^2*e^4 - 1155*a
^3*e^6 + 5*(40*c^3*d^2*e^4 + 99*a*c^2*e^6)*x^4 - 8*(40*c^3*d^3*e^3 + 99*a*c^2*d*e^5)*x^3 + (640*c^3*d^4*e^2 +
1584*a*c^2*d^2*e^4 + 1155*a^2*c*e^6)*x^2 - 4*(640*c^3*d^5*e + 1584*a*c^2*d^3*e^3 + 1155*a^2*c*d*e^5)*x)*sqrt(e
*x + d)/(e^8*x + d*e^7)

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giac [A]  time = 0.18, size = 261, normalized size = 1.32 \begin {gather*} \frac {2}{1155} \, {\left (105 \, {\left (x e + d\right )}^{\frac {11}{2}} c^{3} e^{70} - 770 \, {\left (x e + d\right )}^{\frac {9}{2}} c^{3} d e^{70} + 2475 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} d^{2} e^{70} - 4620 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{3} e^{70} + 5775 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{4} e^{70} - 6930 \, \sqrt {x e + d} c^{3} d^{5} e^{70} + 495 \, {\left (x e + d\right )}^{\frac {7}{2}} a c^{2} e^{72} - 2772 \, {\left (x e + d\right )}^{\frac {5}{2}} a c^{2} d e^{72} + 6930 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} d^{2} e^{72} - 13860 \, \sqrt {x e + d} a c^{2} d^{3} e^{72} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} c e^{74} - 6930 \, \sqrt {x e + d} a^{2} c d e^{74}\right )} e^{\left (-77\right )} - \frac {2 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} e^{\left (-7\right )}}{\sqrt {x e + d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*(105*(x*e + d)^(11/2)*c^3*e^70 - 770*(x*e + d)^(9/2)*c^3*d*e^70 + 2475*(x*e + d)^(7/2)*c^3*d^2*e^70 - 4
620*(x*e + d)^(5/2)*c^3*d^3*e^70 + 5775*(x*e + d)^(3/2)*c^3*d^4*e^70 - 6930*sqrt(x*e + d)*c^3*d^5*e^70 + 495*(
x*e + d)^(7/2)*a*c^2*e^72 - 2772*(x*e + d)^(5/2)*a*c^2*d*e^72 + 6930*(x*e + d)^(3/2)*a*c^2*d^2*e^72 - 13860*sq
rt(x*e + d)*a*c^2*d^3*e^72 + 1155*(x*e + d)^(3/2)*a^2*c*e^74 - 6930*sqrt(x*e + d)*a^2*c*d*e^74)*e^(-77) - 2*(c
^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*e^(-7)/sqrt(x*e + d)

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maple [A]  time = 0.05, size = 205, normalized size = 1.04 \begin {gather*} -\frac {2 \left (-105 c^{3} x^{6} e^{6}+140 c^{3} d \,e^{5} x^{5}-495 a \,c^{2} e^{6} x^{4}-200 c^{3} d^{2} e^{4} x^{4}+792 a \,c^{2} d \,e^{5} x^{3}+320 c^{3} d^{3} e^{3} x^{3}-1155 a^{2} c \,e^{6} x^{2}-1584 a \,c^{2} d^{2} e^{4} x^{2}-640 c^{3} d^{4} e^{2} x^{2}+4620 a^{2} c d \,e^{5} x +6336 a \,c^{2} d^{3} e^{3} x +2560 c^{3} d^{5} e x +1155 e^{6} a^{3}+9240 a^{2} c \,d^{2} e^{4}+12672 a \,c^{2} d^{4} e^{2}+5120 c^{3} d^{6}\right )}{1155 \sqrt {e x +d}\, e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^3/(e*x+d)^(3/2),x)

[Out]

-2/1155/(e*x+d)^(1/2)*(-105*c^3*e^6*x^6+140*c^3*d*e^5*x^5-495*a*c^2*e^6*x^4-200*c^3*d^2*e^4*x^4+792*a*c^2*d*e^
5*x^3+320*c^3*d^3*e^3*x^3-1155*a^2*c*e^6*x^2-1584*a*c^2*d^2*e^4*x^2-640*c^3*d^4*e^2*x^2+4620*a^2*c*d*e^5*x+633
6*a*c^2*d^3*e^3*x+2560*c^3*d^5*e*x+1155*a^3*e^6+9240*a^2*c*d^2*e^4+12672*a*c^2*d^4*e^2+5120*c^3*d^6)/e^7

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maxima [A]  time = 1.32, size = 217, normalized size = 1.10 \begin {gather*} \frac {2 \, {\left (\frac {105 \, {\left (e x + d\right )}^{\frac {11}{2}} c^{3} - 770 \, {\left (e x + d\right )}^{\frac {9}{2}} c^{3} d + 495 \, {\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 924 \, {\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 6930 \, {\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} \sqrt {e x + d}}{e^{6}} - \frac {1155 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}}{\sqrt {e x + d} e^{6}}\right )}}{1155 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*((105*(e*x + d)^(11/2)*c^3 - 770*(e*x + d)^(9/2)*c^3*d + 495*(5*c^3*d^2 + a*c^2*e^2)*(e*x + d)^(7/2) -
924*(5*c^3*d^3 + 3*a*c^2*d*e^2)*(e*x + d)^(5/2) + 1155*(5*c^3*d^4 + 6*a*c^2*d^2*e^2 + a^2*c*e^4)*(e*x + d)^(3/
2) - 6930*(c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*sqrt(e*x + d))/e^6 - 1155*(c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a
^2*c*d^2*e^4 + a^3*e^6)/(sqrt(e*x + d)*e^6))/e

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mupad [B]  time = 0.35, size = 215, normalized size = 1.09 \begin {gather*} \frac {\left (30\,c^3\,d^2+6\,a\,c^2\,e^2\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (6\,a^2\,c\,e^4+36\,a\,c^2\,d^2\,e^2+30\,c^3\,d^4\right )}{3\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}-\frac {\left (40\,c^3\,d^3+24\,a\,c^2\,d\,e^2\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7}-\frac {2\,a^3\,e^6+6\,a^2\,c\,d^2\,e^4+6\,a\,c^2\,d^4\,e^2+2\,c^3\,d^6}{e^7\,\sqrt {d+e\,x}}-\frac {4\,c^3\,d\,{\left (d+e\,x\right )}^{9/2}}{3\,e^7}-\frac {12\,c\,d\,{\left (c\,d^2+a\,e^2\right )}^2\,\sqrt {d+e\,x}}{e^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^3/(d + e*x)^(3/2),x)

[Out]

((30*c^3*d^2 + 6*a*c^2*e^2)*(d + e*x)^(7/2))/(7*e^7) + ((d + e*x)^(3/2)*(30*c^3*d^4 + 6*a^2*c*e^4 + 36*a*c^2*d
^2*e^2))/(3*e^7) + (2*c^3*(d + e*x)^(11/2))/(11*e^7) - ((40*c^3*d^3 + 24*a*c^2*d*e^2)*(d + e*x)^(5/2))/(5*e^7)
- (2*a^3*e^6 + 2*c^3*d^6 + 6*a*c^2*d^4*e^2 + 6*a^2*c*d^2*e^4)/(e^7*(d + e*x)^(1/2)) - (4*c^3*d*(d + e*x)^(9/2
))/(3*e^7) - (12*c*d*(a*e^2 + c*d^2)^2*(d + e*x)^(1/2))/e^7

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sympy [A]  time = 31.81, size = 224, normalized size = 1.13 \begin {gather*} - \frac {4 c^{3} d \left (d + e x\right )^{\frac {9}{2}}}{3 e^{7}} + \frac {2 c^{3} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (6 a c^{2} e^{2} + 30 c^{3} d^{2}\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- 24 a c^{2} d e^{2} - 40 c^{3} d^{3}\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (6 a^{2} c e^{4} + 36 a c^{2} d^{2} e^{2} + 30 c^{3} d^{4}\right )}{3 e^{7}} + \frac {\sqrt {d + e x} \left (- 12 a^{2} c d e^{4} - 24 a c^{2} d^{3} e^{2} - 12 c^{3} d^{5}\right )}{e^{7}} - \frac {2 \left (a e^{2} + c d^{2}\right )^{3}}{e^{7} \sqrt {d + e x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**3/(e*x+d)**(3/2),x)

[Out]

-4*c**3*d*(d + e*x)**(9/2)/(3*e**7) + 2*c**3*(d + e*x)**(11/2)/(11*e**7) + (d + e*x)**(7/2)*(6*a*c**2*e**2 + 3
0*c**3*d**2)/(7*e**7) + (d + e*x)**(5/2)*(-24*a*c**2*d*e**2 - 40*c**3*d**3)/(5*e**7) + (d + e*x)**(3/2)*(6*a**
2*c*e**4 + 36*a*c**2*d**2*e**2 + 30*c**3*d**4)/(3*e**7) + sqrt(d + e*x)*(-12*a**2*c*d*e**4 - 24*a*c**2*d**3*e*
*2 - 12*c**3*d**5)/e**7 - 2*(a*e**2 + c*d**2)**3/(e**7*sqrt(d + e*x))

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