### 3.610 $$\int \frac{(a+c x^2)^3}{(d+e x)^{5/2}} \, dx$$

Optimal. Leaf size=200 $\frac{6 c^2 (d+e x)^{5/2} \left (a e^2+5 c d^2\right )}{5 e^7}-\frac{8 c^2 d (d+e x)^{3/2} \left (3 a e^2+5 c d^2\right )}{3 e^7}+\frac{6 c \sqrt{d+e x} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7}+\frac{12 c d \left (a e^2+c d^2\right )^2}{e^7 \sqrt{d+e x}}-\frac{2 \left (a e^2+c d^2\right )^3}{3 e^7 (d+e x)^{3/2}}+\frac{2 c^3 (d+e x)^{9/2}}{9 e^7}-\frac{12 c^3 d (d+e x)^{7/2}}{7 e^7}$

[Out]

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

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Rubi [A]  time = 0.082786, antiderivative size = 200, 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} $\frac{6 c^2 (d+e x)^{5/2} \left (a e^2+5 c d^2\right )}{5 e^7}-\frac{8 c^2 d (d+e x)^{3/2} \left (3 a e^2+5 c d^2\right )}{3 e^7}+\frac{6 c \sqrt{d+e x} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7}+\frac{12 c d \left (a e^2+c d^2\right )^2}{e^7 \sqrt{d+e x}}-\frac{2 \left (a e^2+c d^2\right )^3}{3 e^7 (d+e x)^{3/2}}+\frac{2 c^3 (d+e x)^{9/2}}{9 e^7}-\frac{12 c^3 d (d+e x)^{7/2}}{7 e^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)^3/(d + e*x)^(5/2),x]

[Out]

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

Mathematica [A]  time = 0.10891, size = 171, normalized size = 0.86 $\frac{2 \left (315 a^2 c e^4 \left (8 d^2+12 d e x+3 e^2 x^2\right )-105 a^3 e^6+63 a c^2 e^2 \left (48 d^2 e^2 x^2+192 d^3 e x+128 d^4-8 d e^3 x^3+3 e^4 x^4\right )+5 c^3 \left (384 d^4 e^2 x^2-64 d^3 e^3 x^3+24 d^2 e^4 x^4+1536 d^5 e x+1024 d^6-12 d e^5 x^5+7 e^6 x^6\right )\right )}{315 e^7 (d+e x)^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + c*x^2)^3/(d + e*x)^(5/2),x]

[Out]

(2*(-105*a^3*e^6 + 315*a^2*c*e^4*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + 63*a*c^2*e^2*(128*d^4 + 192*d^3*e*x + 48*d^2
*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4) + 5*c^3*(1024*d^6 + 1536*d^5*e*x + 384*d^4*e^2*x^2 - 64*d^3*e^3*x^3 + 24*d
^2*e^4*x^4 - 12*d*e^5*x^5 + 7*e^6*x^6)))/(315*e^7*(d + e*x)^(3/2))

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Maple [A]  time = 0.046, size = 205, normalized size = 1. \begin{align*} -{\frac{-70\,{c}^{3}{x}^{6}{e}^{6}+120\,{c}^{3}d{x}^{5}{e}^{5}-378\,a{c}^{2}{e}^{6}{x}^{4}-240\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}+1008\,a{c}^{2}d{e}^{5}{x}^{3}+640\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}-1890\,{a}^{2}c{e}^{6}{x}^{2}-6048\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-3840\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-7560\,{a}^{2}cd{e}^{5}x-24192\,a{c}^{2}{d}^{3}{e}^{3}x-15360\,{c}^{3}{d}^{5}ex+210\,{a}^{3}{e}^{6}-5040\,{a}^{2}c{d}^{2}{e}^{4}-16128\,{d}^{4}{e}^{2}a{c}^{2}-10240\,{c}^{3}{d}^{6}}{315\,{e}^{7}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^3/(e*x+d)^(5/2),x)

[Out]

-2/315/(e*x+d)^(3/2)*(-35*c^3*e^6*x^6+60*c^3*d*e^5*x^5-189*a*c^2*e^6*x^4-120*c^3*d^2*e^4*x^4+504*a*c^2*d*e^5*x
^3+320*c^3*d^3*e^3*x^3-945*a^2*c*e^6*x^2-3024*a*c^2*d^2*e^4*x^2-1920*c^3*d^4*e^2*x^2-3780*a^2*c*d*e^5*x-12096*
a*c^2*d^3*e^3*x-7680*c^3*d^5*e*x+105*a^3*e^6-2520*a^2*c*d^2*e^4-8064*a*c^2*d^4*e^2-5120*c^3*d^6)/e^7

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Maxima [A]  time = 1.60327, size = 290, normalized size = 1.45 \begin{align*} \frac{2 \,{\left (\frac{35 \,{\left (e x + d\right )}^{\frac{9}{2}} c^{3} - 270 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{3} d + 189 \,{\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 420 \,{\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 945 \,{\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} \sqrt{e x + d}}{e^{6}} - \frac{105 \,{\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} - 18 \,{\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{6}}\right )}}{315 \, e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

2/315*((35*(e*x + d)^(9/2)*c^3 - 270*(e*x + d)^(7/2)*c^3*d + 189*(5*c^3*d^2 + a*c^2*e^2)*(e*x + d)^(5/2) - 420
*(5*c^3*d^3 + 3*a*c^2*d*e^2)*(e*x + d)^(3/2) + 945*(5*c^3*d^4 + 6*a*c^2*d^2*e^2 + a^2*c*e^4)*sqrt(e*x + d))/e^
6 - 105*(c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6 - 18*(c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*
(e*x + d))/((e*x + d)^(3/2)*e^6))/e

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Fricas [A]  time = 1.84655, size = 495, normalized size = 2.48 \begin{align*} \frac{2 \,{\left (35 \, c^{3} e^{6} x^{6} - 60 \, c^{3} d e^{5} x^{5} + 5120 \, c^{3} d^{6} + 8064 \, a c^{2} d^{4} e^{2} + 2520 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 3 \,{\left (40 \, c^{3} d^{2} e^{4} + 63 \, a c^{2} e^{6}\right )} x^{4} - 8 \,{\left (40 \, c^{3} d^{3} e^{3} + 63 \, a c^{2} d e^{5}\right )} x^{3} + 3 \,{\left (640 \, c^{3} d^{4} e^{2} + 1008 \, a c^{2} d^{2} e^{4} + 315 \, a^{2} c e^{6}\right )} x^{2} + 12 \,{\left (640 \, c^{3} d^{5} e + 1008 \, a c^{2} d^{3} e^{3} + 315 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{315 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} 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)^(5/2),x, algorithm="fricas")

[Out]

2/315*(35*c^3*e^6*x^6 - 60*c^3*d*e^5*x^5 + 5120*c^3*d^6 + 8064*a*c^2*d^4*e^2 + 2520*a^2*c*d^2*e^4 - 105*a^3*e^
6 + 3*(40*c^3*d^2*e^4 + 63*a*c^2*e^6)*x^4 - 8*(40*c^3*d^3*e^3 + 63*a*c^2*d*e^5)*x^3 + 3*(640*c^3*d^4*e^2 + 100
8*a*c^2*d^2*e^4 + 315*a^2*c*e^6)*x^2 + 12*(640*c^3*d^5*e + 1008*a*c^2*d^3*e^3 + 315*a^2*c*d*e^5)*x)*sqrt(e*x +
d)/(e^9*x^2 + 2*d*e^8*x + d^2*e^7)

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Sympy [A]  time = 31.1395, size = 206, normalized size = 1.03 \begin{align*} - \frac{12 c^{3} d \left (d + e x\right )^{\frac{7}{2}}}{7 e^{7}} + \frac{2 c^{3} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{7}} + \frac{12 c d \left (a e^{2} + c d^{2}\right )^{2}}{e^{7} \sqrt{d + e x}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (6 a c^{2} e^{2} + 30 c^{3} d^{2}\right )}{5 e^{7}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (- 24 a c^{2} d e^{2} - 40 c^{3} d^{3}\right )}{3 e^{7}} + \frac{\sqrt{d + e x} \left (6 a^{2} c e^{4} + 36 a c^{2} d^{2} e^{2} + 30 c^{3} d^{4}\right )}{e^{7}} - \frac{2 \left (a e^{2} + c d^{2}\right )^{3}}{3 e^{7} \left (d + e x\right )^{\frac{3}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**3/(e*x+d)**(5/2),x)

[Out]

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

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Giac [A]  time = 1.40015, size = 344, normalized size = 1.72 \begin{align*} \frac{2}{315} \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} c^{3} e^{56} - 270 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} d e^{56} + 945 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d^{2} e^{56} - 2100 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{3} e^{56} + 4725 \, \sqrt{x e + d} c^{3} d^{4} e^{56} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} a c^{2} e^{58} - 1260 \,{\left (x e + d\right )}^{\frac{3}{2}} a c^{2} d e^{58} + 5670 \, \sqrt{x e + d} a c^{2} d^{2} e^{58} + 945 \, \sqrt{x e + d} a^{2} c e^{60}\right )} e^{\left (-63\right )} + \frac{2 \,{\left (18 \,{\left (x e + d\right )} c^{3} d^{5} - c^{3} d^{6} + 36 \,{\left (x e + d\right )} a c^{2} d^{3} e^{2} - 3 \, a c^{2} d^{4} e^{2} + 18 \,{\left (x e + d\right )} a^{2} c d e^{4} - 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} e^{\left (-7\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/315*(35*(x*e + d)^(9/2)*c^3*e^56 - 270*(x*e + d)^(7/2)*c^3*d*e^56 + 945*(x*e + d)^(5/2)*c^3*d^2*e^56 - 2100*
(x*e + d)^(3/2)*c^3*d^3*e^56 + 4725*sqrt(x*e + d)*c^3*d^4*e^56 + 189*(x*e + d)^(5/2)*a*c^2*e^58 - 1260*(x*e +
d)^(3/2)*a*c^2*d*e^58 + 5670*sqrt(x*e + d)*a*c^2*d^2*e^58 + 945*sqrt(x*e + d)*a^2*c*e^60)*e^(-63) + 2/3*(18*(x
*e + d)*c^3*d^5 - c^3*d^6 + 36*(x*e + d)*a*c^2*d^3*e^2 - 3*a*c^2*d^4*e^2 + 18*(x*e + d)*a^2*c*d*e^4 - 3*a^2*c*
d^2*e^4 - a^3*e^6)*e^(-7)/(x*e + d)^(3/2)