3.1437 \(\int \frac{(A+B x) (a+c x^2)^2}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=214 \[ \frac{4 c (d+e x)^{5/2} \left (a B e^2-2 A c d e+5 B c d^2\right )}{5 e^6}-\frac{4 c (d+e x)^{3/2} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6}+\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^6}+\frac{2 \left (a e^2+c d^2\right )^2 (B d-A e)}{e^6 \sqrt{d+e x}}-\frac{2 c^2 (d+e x)^{7/2} (5 B d-A e)}{7 e^6}+\frac{2 B c^2 (d+e x)^{9/2}}{9 e^6} \]

[Out]

(2*(B*d - A*e)*(c*d^2 + a*e^2)^2)/(e^6*Sqrt[d + e*x]) + (2*(c*d^2 + a*e^2)*(5*B*c*d^2 - 4*A*c*d*e + a*B*e^2)*S
qrt[d + e*x])/e^6 - (4*c*(5*B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^(3/2))/(3*e^6) + (4*c*(5*
B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(5/2))/(5*e^6) - (2*c^2*(5*B*d - A*e)*(d + e*x)^(7/2))/(7*e^6) + (2*B
*c^2*(d + e*x)^(9/2))/(9*e^6)

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Rubi [A]  time = 0.0934932, antiderivative size = 214, normalized size of antiderivative = 1., 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} \[ \frac{4 c (d+e x)^{5/2} \left (a B e^2-2 A c d e+5 B c d^2\right )}{5 e^6}-\frac{4 c (d+e x)^{3/2} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6}+\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^6}+\frac{2 \left (a e^2+c d^2\right )^2 (B d-A e)}{e^6 \sqrt{d+e x}}-\frac{2 c^2 (d+e x)^{7/2} (5 B d-A e)}{7 e^6}+\frac{2 B c^2 (d+e x)^{9/2}}{9 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.156, size = 214, normalized size = 1. \[ \frac{2 B \left (315 a^2 e^4 (2 d+e x)+126 a c e^2 \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )+5 c^2 \left (-32 d^3 e^2 x^2+16 d^2 e^3 x^3+128 d^4 e x+256 d^5-10 d e^4 x^4+7 e^5 x^5\right )\right )-6 A e \left (105 a^2 e^4+70 a c e^2 \left (8 d^2+4 d e x-e^2 x^2\right )+3 c^2 \left (-16 d^2 e^2 x^2+64 d^3 e x+128 d^4+8 d e^3 x^3-5 e^4 x^4\right )\right )}{315 e^6 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*A*e*(105*a^2*e^4 + 70*a*c*e^2*(8*d^2 + 4*d*e*x - e^2*x^2) + 3*c^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)) + 2*B*(315*a^2*e^4*(2*d + e*x) + 126*a*c*e^2*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^
3*x^3) + 5*c^2*(256*d^5 + 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^4 + 7*e^5*x^5)))/(315*e^6
*Sqrt[d + e*x])

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Maple [A]  time = 0.006, size = 259, normalized size = 1.2 \begin{align*} -{\frac{-70\,B{c}^{2}{x}^{5}{e}^{5}-90\,A{c}^{2}{e}^{5}{x}^{4}+100\,B{c}^{2}d{e}^{4}{x}^{4}+144\,A{c}^{2}d{e}^{4}{x}^{3}-252\,Bac{e}^{5}{x}^{3}-160\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}-420\,Aac{e}^{5}{x}^{2}-288\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}+504\,Bacd{e}^{4}{x}^{2}+320\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}+1680\,Aacd{e}^{4}x+1152\,A{c}^{2}{d}^{3}{e}^{2}x-630\,B{a}^{2}{e}^{5}x-2016\,Bac{d}^{2}{e}^{3}x-1280\,B{c}^{2}{d}^{4}ex+630\,A{a}^{2}{e}^{5}+3360\,A{d}^{2}ac{e}^{3}+2304\,A{d}^{4}{c}^{2}e-1260\,B{a}^{2}d{e}^{4}-4032\,aBc{d}^{3}{e}^{2}-2560\,B{c}^{2}{d}^{5}}{315\,{e}^{6}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315/(e*x+d)^(1/2)*(-35*B*c^2*e^5*x^5-45*A*c^2*e^5*x^4+50*B*c^2*d*e^4*x^4+72*A*c^2*d*e^4*x^3-126*B*a*c*e^5*x
^3-80*B*c^2*d^2*e^3*x^3-210*A*a*c*e^5*x^2-144*A*c^2*d^2*e^3*x^2+252*B*a*c*d*e^4*x^2+160*B*c^2*d^3*e^2*x^2+840*
A*a*c*d*e^4*x+576*A*c^2*d^3*e^2*x-315*B*a^2*e^5*x-1008*B*a*c*d^2*e^3*x-640*B*c^2*d^4*e*x+315*A*a^2*e^5+1680*A*
a*c*d^2*e^3+1152*A*c^2*d^4*e-630*B*a^2*d*e^4-2016*B*a*c*d^3*e^2-1280*B*c^2*d^5)/e^6

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Maxima [A]  time = 1.02132, size = 346, normalized size = 1.62 \begin{align*} \frac{2 \,{\left (\frac{35 \,{\left (e x + d\right )}^{\frac{9}{2}} B c^{2} - 45 \,{\left (5 \, B c^{2} d - A c^{2} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 126 \,{\left (5 \, B c^{2} d^{2} - 2 \, A c^{2} d e + B a c e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 210 \,{\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 )}^{\frac{3}{2}} + 315 \,{\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 )} \sqrt{e x + d}}{e^{5}} + \frac{315 \,{\left (B c^{2} d^{5} - A c^{2} d^{4} e + 2 \, B a c d^{3} e^{2} - 2 \, A a c d^{2} e^{3} + B a^{2} d e^{4} - A a^{2} e^{5}\right )}}{\sqrt{e x + d} e^{5}}\right )}}{315 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*((35*(e*x + d)^(9/2)*B*c^2 - 45*(5*B*c^2*d - A*c^2*e)*(e*x + d)^(7/2) + 126*(5*B*c^2*d^2 - 2*A*c^2*d*e +
B*a*c*e^2)*(e*x + d)^(5/2) - 210*(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)^(3/2) +
315*(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)*sqrt(e*x + d))/e^5 + 315*(B*c^
2*d^5 - A*c^2*d^4*e + 2*B*a*c*d^3*e^2 - 2*A*a*c*d^2*e^3 + B*a^2*d*e^4 - A*a^2*e^5)/(sqrt(e*x + d)*e^5))/e

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Fricas [A]  time = 1.51797, size = 598, normalized size = 2.79 \begin{align*} \frac{2 \,{\left (35 \, B c^{2} e^{5} x^{5} + 1280 \, B c^{2} d^{5} - 1152 \, A c^{2} d^{4} e + 2016 \, B a c d^{3} e^{2} - 1680 \, A a c d^{2} e^{3} + 630 \, B a^{2} d e^{4} - 315 \, A a^{2} e^{5} - 5 \,{\left (10 \, B c^{2} d e^{4} - 9 \, A c^{2} e^{5}\right )} x^{4} + 2 \,{\left (40 \, B c^{2} d^{2} e^{3} - 36 \, A c^{2} d e^{4} + 63 \, B a c e^{5}\right )} x^{3} - 2 \,{\left (80 \, B c^{2} d^{3} e^{2} - 72 \, A c^{2} d^{2} e^{3} + 126 \, B a c d e^{4} - 105 \, A a c e^{5}\right )} x^{2} +{\left (640 \, B c^{2} d^{4} e - 576 \, A c^{2} d^{3} e^{2} + 1008 \, B a c d^{2} e^{3} - 840 \, A a c d e^{4} + 315 \, B a^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}{315 \,{\left (e^{7} x + d e^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*B*c^2*e^5*x^5 + 1280*B*c^2*d^5 - 1152*A*c^2*d^4*e + 2016*B*a*c*d^3*e^2 - 1680*A*a*c*d^2*e^3 + 630*B*
a^2*d*e^4 - 315*A*a^2*e^5 - 5*(10*B*c^2*d*e^4 - 9*A*c^2*e^5)*x^4 + 2*(40*B*c^2*d^2*e^3 - 36*A*c^2*d*e^4 + 63*B
*a*c*e^5)*x^3 - 2*(80*B*c^2*d^3*e^2 - 72*A*c^2*d^2*e^3 + 126*B*a*c*d*e^4 - 105*A*a*c*e^5)*x^2 + (640*B*c^2*d^4
*e - 576*A*c^2*d^3*e^2 + 1008*B*a*c*d^2*e^3 - 840*A*a*c*d*e^4 + 315*B*a^2*e^5)*x)*sqrt(e*x + d)/(e^7*x + d*e^6
)

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Sympy [A]  time = 35.2827, size = 253, normalized size = 1.18 \begin{align*} \frac{2 B c^{2} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{6}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (2 A c^{2} e - 10 B c^{2} d\right )}{7 e^{6}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (- 8 A c^{2} d e + 4 B a c e^{2} + 20 B c^{2} d^{2}\right )}{5 e^{6}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (4 A a c e^{3} + 12 A c^{2} d^{2} e - 12 B a c d e^{2} - 20 B c^{2} d^{3}\right )}{3 e^{6}} + \frac{\sqrt{d + e x} \left (- 8 A a c d e^{3} - 8 A c^{2} d^{3} e + 2 B a^{2} e^{4} + 12 B a c d^{2} e^{2} + 10 B c^{2} d^{4}\right )}{e^{6}} + \frac{2 \left (- A e + B d\right ) \left (a e^{2} + c d^{2}\right )^{2}}{e^{6} \sqrt{d + e x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*B*c**2*(d + e*x)**(9/2)/(9*e**6) + (d + e*x)**(7/2)*(2*A*c**2*e - 10*B*c**2*d)/(7*e**6) + (d + e*x)**(5/2)*(
-8*A*c**2*d*e + 4*B*a*c*e**2 + 20*B*c**2*d**2)/(5*e**6) + (d + e*x)**(3/2)*(4*A*a*c*e**3 + 12*A*c**2*d**2*e -
12*B*a*c*d*e**2 - 20*B*c**2*d**3)/(3*e**6) + sqrt(d + e*x)*(-8*A*a*c*d*e**3 - 8*A*c**2*d**3*e + 2*B*a**2*e**4
+ 12*B*a*c*d**2*e**2 + 10*B*c**2*d**4)/e**6 + 2*(-A*e + B*d)*(a*e**2 + c*d**2)**2/(e**6*sqrt(d + e*x))

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Giac [A]  time = 1.2252, size = 447, normalized size = 2.09 \begin{align*} \frac{2}{315} \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} B c^{2} e^{48} - 225 \,{\left (x e + d\right )}^{\frac{7}{2}} B c^{2} d e^{48} + 630 \,{\left (x e + d\right )}^{\frac{5}{2}} B c^{2} d^{2} e^{48} - 1050 \,{\left (x e + d\right )}^{\frac{3}{2}} B c^{2} d^{3} e^{48} + 1575 \, \sqrt{x e + d} B c^{2} d^{4} e^{48} + 45 \,{\left (x e + d\right )}^{\frac{7}{2}} A c^{2} e^{49} - 252 \,{\left (x e + d\right )}^{\frac{5}{2}} A c^{2} d e^{49} + 630 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{2} d^{2} e^{49} - 1260 \, \sqrt{x e + d} A c^{2} d^{3} e^{49} + 126 \,{\left (x e + d\right )}^{\frac{5}{2}} B a c e^{50} - 630 \,{\left (x e + d\right )}^{\frac{3}{2}} B a c d e^{50} + 1890 \, \sqrt{x e + d} B a c d^{2} e^{50} + 210 \,{\left (x e + d\right )}^{\frac{3}{2}} A a c e^{51} - 1260 \, \sqrt{x e + d} A a c d e^{51} + 315 \, \sqrt{x e + d} B a^{2} e^{52}\right )} e^{\left (-54\right )} + \frac{2 \,{\left (B c^{2} d^{5} - A c^{2} d^{4} e + 2 \, B a c d^{3} e^{2} - 2 \, A a c d^{2} e^{3} + B a^{2} d e^{4} - A a^{2} e^{5}\right )} e^{\left (-6\right )}}{\sqrt{x e + d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*(x*e + d)^(9/2)*B*c^2*e^48 - 225*(x*e + d)^(7/2)*B*c^2*d*e^48 + 630*(x*e + d)^(5/2)*B*c^2*d^2*e^48 -
1050*(x*e + d)^(3/2)*B*c^2*d^3*e^48 + 1575*sqrt(x*e + d)*B*c^2*d^4*e^48 + 45*(x*e + d)^(7/2)*A*c^2*e^49 - 252
*(x*e + d)^(5/2)*A*c^2*d*e^49 + 630*(x*e + d)^(3/2)*A*c^2*d^2*e^49 - 1260*sqrt(x*e + d)*A*c^2*d^3*e^49 + 126*(
x*e + d)^(5/2)*B*a*c*e^50 - 630*(x*e + d)^(3/2)*B*a*c*d*e^50 + 1890*sqrt(x*e + d)*B*a*c*d^2*e^50 + 210*(x*e +
d)^(3/2)*A*a*c*e^51 - 1260*sqrt(x*e + d)*A*a*c*d*e^51 + 315*sqrt(x*e + d)*B*a^2*e^52)*e^(-54) + 2*(B*c^2*d^5 -
A*c^2*d^4*e + 2*B*a*c*d^3*e^2 - 2*A*a*c*d^2*e^3 + B*a^2*d*e^4 - A*a^2*e^5)*e^(-6)/sqrt(x*e + d)