### 3.611 $$\int \frac{(a+c x^2)^3}{(d+e x)^{7/2}} \, dx$$

Optimal. Leaf size=196 $\frac{2 c^2 (d+e x)^{3/2} \left (a e^2+5 c d^2\right )}{e^7}-\frac{8 c^2 d \sqrt{d+e x} \left (3 a e^2+5 c d^2\right )}{e^7}-\frac{6 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 \sqrt{d+e x}}+\frac{4 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac{2 \left (a e^2+c d^2\right )^3}{5 e^7 (d+e x)^{5/2}}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7}-\frac{12 c^3 d (d+e x)^{5/2}}{5 e^7}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0807837, antiderivative size = 196, 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} $\frac{2 c^2 (d+e x)^{3/2} \left (a e^2+5 c d^2\right )}{e^7}-\frac{8 c^2 d \sqrt{d+e x} \left (3 a e^2+5 c d^2\right )}{e^7}-\frac{6 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 \sqrt{d+e x}}+\frac{4 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac{2 \left (a e^2+c d^2\right )^3}{5 e^7 (d+e x)^{5/2}}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7}-\frac{12 c^3 d (d+e x)^{5/2}}{5 e^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)^3/(d + e*x)^(7/2),x]

[Out]

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

Mathematica [A]  time = 0.115347, size = 170, normalized size = 0.87 $-\frac{2 \left (7 a^2 c e^4 \left (8 d^2+20 d e x+15 e^2 x^2\right )+7 a^3 e^6+7 a c^2 e^2 \left (240 d^2 e^2 x^2+320 d^3 e x+128 d^4+40 d e^3 x^3-5 e^4 x^4\right )+c^3 \left (1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+2560 d^5 e x+1024 d^6+12 d e^5 x^5-5 e^6 x^6\right )\right )}{35 e^7 (d+e x)^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*(7*a^3*e^6 + 7*a^2*c*e^4*(8*d^2 + 20*d*e*x + 15*e^2*x^2) + 7*a*c^2*e^2*(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) + c^3*(1024*d^6 + 2560*d^5*e*x + 1920*d^4*e^2*x^2 + 320*d^3*e^3*x^3 - 40*d^2
*e^4*x^4 + 12*d*e^5*x^5 - 5*e^6*x^6)))/(35*e^7*(d + e*x)^(5/2))

________________________________________________________________________________________

Maple [A]  time = 0.044, size = 205, normalized size = 1.1 \begin{align*} -{\frac{-10\,{c}^{3}{x}^{6}{e}^{6}+24\,{c}^{3}d{x}^{5}{e}^{5}-70\,a{c}^{2}{e}^{6}{x}^{4}-80\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}+560\,a{c}^{2}d{e}^{5}{x}^{3}+640\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+210\,{a}^{2}c{e}^{6}{x}^{2}+3360\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}+3840\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+280\,{a}^{2}cd{e}^{5}x+4480\,a{c}^{2}{d}^{3}{e}^{3}x+5120\,{c}^{3}{d}^{5}ex+14\,{a}^{3}{e}^{6}+112\,{a}^{2}c{d}^{2}{e}^{4}+1792\,{d}^{4}{e}^{2}a{c}^{2}+2048\,{c}^{3}{d}^{6}}{35\,{e}^{7}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35/(e*x+d)^(5/2)*(-5*c^3*e^6*x^6+12*c^3*d*e^5*x^5-35*a*c^2*e^6*x^4-40*c^3*d^2*e^4*x^4+280*a*c^2*d*e^5*x^3+3
20*c^3*d^3*e^3*x^3+105*a^2*c*e^6*x^2+1680*a*c^2*d^2*e^4*x^2+1920*c^3*d^4*e^2*x^2+140*a^2*c*d*e^5*x+2240*a*c^2*
d^3*e^3*x+2560*c^3*d^5*e*x+7*a^3*e^6+56*a^2*c*d^2*e^4+896*a*c^2*d^4*e^2+1024*c^3*d^6)/e^7

________________________________________________________________________________________

Maxima [A]  time = 1.61279, size = 290, normalized size = 1.48 \begin{align*} \frac{2 \,{\left (\frac{5 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{3} - 42 \,{\left (e x + d\right )}^{\frac{5}{2}} c^{3} d + 35 \,{\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 140 \,{\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} \sqrt{e x + d}}{e^{6}} - \frac{7 \,{\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} + 15 \,{\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 )}^{2} - 10 \,{\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{5}{2}} e^{6}}\right )}}{35 \, e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*((5*(e*x + d)^(7/2)*c^3 - 42*(e*x + d)^(5/2)*c^3*d + 35*(5*c^3*d^2 + a*c^2*e^2)*(e*x + d)^(3/2) - 140*(5*
c^3*d^3 + 3*a*c^2*d*e^2)*sqrt(e*x + d))/e^6 - 7*(c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6 + 15*(5
*c^3*d^4 + 6*a*c^2*d^2*e^2 + a^2*c*e^4)*(e*x + d)^2 - 10*(c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*(e*x + d))/
((e*x + d)^(5/2)*e^6))/e

________________________________________________________________________________________

Fricas [A]  time = 1.76616, size = 498, normalized size = 2.54 \begin{align*} \frac{2 \,{\left (5 \, c^{3} e^{6} x^{6} - 12 \, c^{3} d e^{5} x^{5} - 1024 \, c^{3} d^{6} - 896 \, a c^{2} d^{4} e^{2} - 56 \, a^{2} c d^{2} e^{4} - 7 \, a^{3} e^{6} + 5 \,{\left (8 \, c^{3} d^{2} e^{4} + 7 \, a c^{2} e^{6}\right )} x^{4} - 40 \,{\left (8 \, c^{3} d^{3} e^{3} + 7 \, a c^{2} d e^{5}\right )} x^{3} - 15 \,{\left (128 \, c^{3} d^{4} e^{2} + 112 \, a c^{2} d^{2} e^{4} + 7 \, a^{2} c e^{6}\right )} x^{2} - 20 \,{\left (128 \, c^{3} d^{5} e + 112 \, a c^{2} d^{3} e^{3} + 7 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{35 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} 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)^(7/2),x, algorithm="fricas")

[Out]

2/35*(5*c^3*e^6*x^6 - 12*c^3*d*e^5*x^5 - 1024*c^3*d^6 - 896*a*c^2*d^4*e^2 - 56*a^2*c*d^2*e^4 - 7*a^3*e^6 + 5*(
8*c^3*d^2*e^4 + 7*a*c^2*e^6)*x^4 - 40*(8*c^3*d^3*e^3 + 7*a*c^2*d*e^5)*x^3 - 15*(128*c^3*d^4*e^2 + 112*a*c^2*d^
2*e^4 + 7*a^2*c*e^6)*x^2 - 20*(128*c^3*d^5*e + 112*a*c^2*d^3*e^3 + 7*a^2*c*d*e^5)*x)*sqrt(e*x + d)/(e^10*x^3 +
3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)

________________________________________________________________________________________

Sympy [A]  time = 48.5068, size = 197, normalized size = 1.01 \begin{align*} - \frac{12 c^{3} d \left (d + e x\right )^{\frac{5}{2}}}{5 e^{7}} + \frac{2 c^{3} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{7}} + \frac{4 c d \left (a e^{2} + c d^{2}\right )^{2}}{e^{7} \left (d + e x\right )^{\frac{3}{2}}} - \frac{6 c \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{e^{7} \sqrt{d + e x}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (6 a c^{2} e^{2} + 30 c^{3} d^{2}\right )}{3 e^{7}} + \frac{\sqrt{d + e x} \left (- 24 a c^{2} d e^{2} - 40 c^{3} d^{3}\right )}{e^{7}} - \frac{2 \left (a e^{2} + c d^{2}\right )^{3}}{5 e^{7} \left (d + e x\right )^{\frac{5}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.34764, size = 339, normalized size = 1.73 \begin{align*} \frac{2}{35} \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} e^{42} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d e^{42} + 175 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{2} e^{42} - 700 \, \sqrt{x e + d} c^{3} d^{3} e^{42} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a c^{2} e^{44} - 420 \, \sqrt{x e + d} a c^{2} d e^{44}\right )} e^{\left (-49\right )} - \frac{2 \,{\left (75 \,{\left (x e + d\right )}^{2} c^{3} d^{4} - 10 \,{\left (x e + d\right )} c^{3} d^{5} + c^{3} d^{6} + 90 \,{\left (x e + d\right )}^{2} a c^{2} d^{2} e^{2} - 20 \,{\left (x e + d\right )} a c^{2} d^{3} e^{2} + 3 \, a c^{2} d^{4} e^{2} + 15 \,{\left (x e + d\right )}^{2} a^{2} c e^{4} - 10 \,{\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 )}}{5 \,{\left (x e + d\right )}^{\frac{5}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*(x*e + d)^(7/2)*c^3*e^42 - 42*(x*e + d)^(5/2)*c^3*d*e^42 + 175*(x*e + d)^(3/2)*c^3*d^2*e^42 - 700*sqrt
(x*e + d)*c^3*d^3*e^42 + 35*(x*e + d)^(3/2)*a*c^2*e^44 - 420*sqrt(x*e + d)*a*c^2*d*e^44)*e^(-49) - 2/5*(75*(x*
e + d)^2*c^3*d^4 - 10*(x*e + d)*c^3*d^5 + c^3*d^6 + 90*(x*e + d)^2*a*c^2*d^2*e^2 - 20*(x*e + d)*a*c^2*d^3*e^2
+ 3*a*c^2*d^4*e^2 + 15*(x*e + d)^2*a^2*c*e^4 - 10*(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)^(5/2)