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

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

[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
)

________________________________________________________________________________________

Rubi [A]  time = 0.0834626, antiderivative size = 198, normalized size of antiderivative = 1., 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} \[ \frac{6 c^2 (d+e x)^{7/2} \left (a e^2+5 c d^2\right )}{7 e^7}-\frac{8 c^2 d (d+e x)^{5/2} \left (3 a e^2+5 c d^2\right )}{5 e^7}+\frac{2 c (d+e x)^{3/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7}-\frac{12 c d \sqrt{d+e x} \left (a e^2+c d^2\right )^2}{e^7}-\frac{2 \left (a e^2+c d^2\right )^3}{e^7 \sqrt{d+e x}}+\frac{2 c^3 (d+e x)^{11/2}}{11 e^7}-\frac{4 c^3 d (d+e x)^{9/2}}{3 e^7} \]

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*}

Mathematica [A]  time = 0.115819, size = 171, normalized size = 0.86 \[ -\frac{2 \left (1155 a^2 c e^4 \left (8 d^2+4 d e x-e^2 x^2\right )+1155 a^3 e^6+99 a c^2 e^2 \left (-16 d^2 e^2 x^2+64 d^3 e x+128 d^4+8 d e^3 x^3-5 e^4 x^4\right )+5 c^3 \left (-128 d^4 e^2 x^2+64 d^3 e^3 x^3-40 d^2 e^4 x^4+512 d^5 e x+1024 d^6+28 d e^5 x^5-21 e^6 x^6\right )\right )}{1155 e^7 \sqrt{d+e x}} \]

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])

________________________________________________________________________________________

Maple [A]  time = 0.045, size = 205, normalized size = 1. \begin{align*} -{\frac{-210\,{c}^{3}{x}^{6}{e}^{6}+280\,{c}^{3}d{x}^{5}{e}^{5}-990\,a{c}^{2}{e}^{6}{x}^{4}-400\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}+1584\,a{c}^{2}d{e}^{5}{x}^{3}+640\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}-2310\,{a}^{2}c{e}^{6}{x}^{2}-3168\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-1280\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+9240\,{a}^{2}cd{e}^{5}x+12672\,a{c}^{2}{d}^{3}{e}^{3}x+5120\,{c}^{3}{d}^{5}ex+2310\,{a}^{3}{e}^{6}+18480\,{a}^{2}c{d}^{2}{e}^{4}+25344\,{d}^{4}{e}^{2}a{c}^{2}+10240\,{c}^{3}{d}^{6}}{1155\,{e}^{7}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}

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

________________________________________________________________________________________

Maxima [A]  time = 1.53718, size = 293, normalized size = 1.48 \begin{align*} \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{align*}

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

________________________________________________________________________________________

Fricas [A]  time = 1.87235, size = 479, normalized size = 2.42 \begin{align*} \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{align*}

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)

________________________________________________________________________________________

Sympy [A]  time = 23.6801, size = 224, normalized size = 1.13 \begin{align*} - \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{align*}

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))

________________________________________________________________________________________

Giac [A]  time = 1.4838, size = 352, normalized size = 1.78 \begin{align*} \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{align*}

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)