3.1438 \(\int \frac{(A+B x) (a+c x^2)^2}{(d+e x)^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0917347, 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} \[ \frac{4 c (d+e x)^{3/2} \left (a B e^2-2 A c d e+5 B c d^2\right )}{3 e^6}-\frac{4 c \sqrt{d+e x} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6}-\frac{2 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^6 \sqrt{d+e x}}+\frac{2 \left (a e^2+c d^2\right )^2 (B d-A e)}{3 e^6 (d+e x)^{3/2}}-\frac{2 c^2 (d+e x)^{5/2} (5 B d-A e)}{5 e^6}+\frac{2 B c^2 (d+e x)^{7/2}}{7 e^6} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^(5/2),x]

[Out]

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

Mathematica [A]  time = 0.15249, size = 214, normalized size = 1. \[ \frac{14 A e \left (-5 a^2 e^4+10 a c e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+c^2 \left (48 d^2 e^2 x^2+192 d^3 e x+128 d^4-8 d e^3 x^3+3 e^4 x^4\right )\right )-10 B \left (7 a^2 e^4 (2 d+3 e x)+14 a c e^2 \left (24 d^2 e x+16 d^3+6 d e^2 x^2-e^3 x^3\right )+c^2 \left (96 d^3 e^2 x^2-16 d^2 e^3 x^3+384 d^4 e x+256 d^5+6 d e^4 x^4-3 e^5 x^5\right )\right )}{105 e^6 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(14*A*e*(-5*a^2*e^4 + 10*a*c*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + c^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)) - 10*B*(7*a^2*e^4*(2*d + 3*e*x) + 14*a*c*e^2*(16*d^3 + 24*d^2*e*x + 6*d*e^2*x^2 -
e^3*x^3) + c^2*(256*d^5 + 384*d^4*e*x + 96*d^3*e^2*x^2 - 16*d^2*e^3*x^3 + 6*d*e^4*x^4 - 3*e^5*x^5)))/(105*e^6*
(d + e*x)^(3/2))

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Maple [A]  time = 0.006, size = 259, normalized size = 1.2 \begin{align*} -{\frac{-30\,B{c}^{2}{x}^{5}{e}^{5}-42\,A{c}^{2}{e}^{5}{x}^{4}+60\,B{c}^{2}d{e}^{4}{x}^{4}+112\,A{c}^{2}d{e}^{4}{x}^{3}-140\,Bac{e}^{5}{x}^{3}-160\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}-420\,Aac{e}^{5}{x}^{2}-672\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}+840\,Bacd{e}^{4}{x}^{2}+960\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}-1680\,Aacd{e}^{4}x-2688\,A{c}^{2}{d}^{3}{e}^{2}x+210\,B{a}^{2}{e}^{5}x+3360\,Bac{d}^{2}{e}^{3}x+3840\,B{c}^{2}{d}^{4}ex+70\,A{a}^{2}{e}^{5}-1120\,A{d}^{2}ac{e}^{3}-1792\,A{d}^{4}{c}^{2}e+140\,B{a}^{2}d{e}^{4}+2240\,aBc{d}^{3}{e}^{2}+2560\,B{c}^{2}{d}^{5}}{105\,{e}^{6}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \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)^(5/2),x)

[Out]

-2/105/(e*x+d)^(3/2)*(-15*B*c^2*e^5*x^5-21*A*c^2*e^5*x^4+30*B*c^2*d*e^4*x^4+56*A*c^2*d*e^4*x^3-70*B*a*c*e^5*x^
3-80*B*c^2*d^2*e^3*x^3-210*A*a*c*e^5*x^2-336*A*c^2*d^2*e^3*x^2+420*B*a*c*d*e^4*x^2+480*B*c^2*d^3*e^2*x^2-840*A
*a*c*d*e^4*x-1344*A*c^2*d^3*e^2*x+105*B*a^2*e^5*x+1680*B*a*c*d^2*e^3*x+1920*B*c^2*d^4*e*x+35*A*a^2*e^5-560*A*a
*c*d^2*e^3-896*A*c^2*d^4*e+70*B*a^2*d*e^4+1120*B*a*c*d^3*e^2+1280*B*c^2*d^5)/e^6

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Maxima [A]  time = 1.0017, size = 343, normalized size = 1.6 \begin{align*} \frac{2 \,{\left (\frac{15 \,{\left (e x + d\right )}^{\frac{7}{2}} B c^{2} - 21 \,{\left (5 \, B c^{2} d - A c^{2} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 70 \,{\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{3}{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 )} \sqrt{e x + d}}{e^{5}} + \frac{35 \,{\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} - 3 \,{\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 )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{5}}\right )}}{105 \, 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)^(5/2),x, algorithm="maxima")

[Out]

2/105*((15*(e*x + d)^(7/2)*B*c^2 - 21*(5*B*c^2*d - A*c^2*e)*(e*x + d)^(5/2) + 70*(5*B*c^2*d^2 - 2*A*c^2*d*e +
B*a*c*e^2)*(e*x + d)^(3/2) - 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)*sqrt(e*x + d))/e^5
+ 35*(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 - 3*(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)*(e*x + d))/((e*x + d)^(3/2)*e^5))/e

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Fricas [A]  time = 1.55039, size = 612, normalized size = 2.86 \begin{align*} \frac{2 \,{\left (15 \, B c^{2} e^{5} x^{5} - 1280 \, B c^{2} d^{5} + 896 \, A c^{2} d^{4} e - 1120 \, B a c d^{3} e^{2} + 560 \, A a c d^{2} e^{3} - 70 \, B a^{2} d e^{4} - 35 \, A a^{2} e^{5} - 3 \,{\left (10 \, B c^{2} d e^{4} - 7 \, A c^{2} e^{5}\right )} x^{4} + 2 \,{\left (40 \, B c^{2} d^{2} e^{3} - 28 \, A c^{2} d e^{4} + 35 \, B a c e^{5}\right )} x^{3} - 6 \,{\left (80 \, B c^{2} d^{3} e^{2} - 56 \, A c^{2} d^{2} e^{3} + 70 \, B a c d e^{4} - 35 \, A a c e^{5}\right )} x^{2} - 3 \,{\left (640 \, B c^{2} d^{4} e - 448 \, A c^{2} d^{3} e^{2} + 560 \, B a c d^{2} e^{3} - 280 \, A a c d e^{4} + 35 \, B a^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}{105 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} 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)^(5/2),x, algorithm="fricas")

[Out]

2/105*(15*B*c^2*e^5*x^5 - 1280*B*c^2*d^5 + 896*A*c^2*d^4*e - 1120*B*a*c*d^3*e^2 + 560*A*a*c*d^2*e^3 - 70*B*a^2
*d*e^4 - 35*A*a^2*e^5 - 3*(10*B*c^2*d*e^4 - 7*A*c^2*e^5)*x^4 + 2*(40*B*c^2*d^2*e^3 - 28*A*c^2*d*e^4 + 35*B*a*c
*e^5)*x^3 - 6*(80*B*c^2*d^3*e^2 - 56*A*c^2*d^2*e^3 + 70*B*a*c*d*e^4 - 35*A*a*c*e^5)*x^2 - 3*(640*B*c^2*d^4*e -
448*A*c^2*d^3*e^2 + 560*B*a*c*d^2*e^3 - 280*A*a*c*d*e^4 + 35*B*a^2*e^5)*x)*sqrt(e*x + d)/(e^8*x^2 + 2*d*e^7*x
+ d^2*e^6)

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Sympy [A]  time = 44.7661, size = 231, normalized size = 1.08 \begin{align*} \frac{2 B c^{2} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{6}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (2 A c^{2} e - 10 B c^{2} d\right )}{5 e^{6}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (- 8 A c^{2} d e + 4 B a c e^{2} + 20 B c^{2} d^{2}\right )}{3 e^{6}} + \frac{\sqrt{d + e x} \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 )}{e^{6}} - \frac{2 \left (a e^{2} + c d^{2}\right ) \left (- 4 A c d e + B a e^{2} + 5 B c d^{2}\right )}{e^{6} \sqrt{d + e x}} + \frac{2 \left (- A e + B d\right ) \left (a e^{2} + c d^{2}\right )^{2}}{3 e^{6} \left (d + e x\right )^{\frac{3}{2}}} \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)**(5/2),x)

[Out]

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

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Giac [A]  time = 1.26671, size = 431, normalized size = 2.01 \begin{align*} \frac{2}{105} \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} B c^{2} e^{36} - 105 \,{\left (x e + d\right )}^{\frac{5}{2}} B c^{2} d e^{36} + 350 \,{\left (x e + d\right )}^{\frac{3}{2}} B c^{2} d^{2} e^{36} - 1050 \, \sqrt{x e + d} B c^{2} d^{3} e^{36} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} A c^{2} e^{37} - 140 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{2} d e^{37} + 630 \, \sqrt{x e + d} A c^{2} d^{2} e^{37} + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} B a c e^{38} - 630 \, \sqrt{x e + d} B a c d e^{38} + 210 \, \sqrt{x e + d} A a c e^{39}\right )} e^{\left (-42\right )} - \frac{2 \,{\left (15 \,{\left (x e + d\right )} B c^{2} d^{4} - B c^{2} d^{5} - 12 \,{\left (x e + d\right )} A c^{2} d^{3} e + A c^{2} d^{4} e + 18 \,{\left (x e + d\right )} B a c d^{2} e^{2} - 2 \, B a c d^{3} e^{2} - 12 \,{\left (x e + d\right )} A a c d e^{3} + 2 \, A a c d^{2} e^{3} + 3 \,{\left (x e + d\right )} B a^{2} e^{4} - B a^{2} d e^{4} + A a^{2} e^{5}\right )} e^{\left (-6\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \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)^(5/2),x, algorithm="giac")

[Out]

2/105*(15*(x*e + d)^(7/2)*B*c^2*e^36 - 105*(x*e + d)^(5/2)*B*c^2*d*e^36 + 350*(x*e + d)^(3/2)*B*c^2*d^2*e^36 -
1050*sqrt(x*e + d)*B*c^2*d^3*e^36 + 21*(x*e + d)^(5/2)*A*c^2*e^37 - 140*(x*e + d)^(3/2)*A*c^2*d*e^37 + 630*sq
rt(x*e + d)*A*c^2*d^2*e^37 + 70*(x*e + d)^(3/2)*B*a*c*e^38 - 630*sqrt(x*e + d)*B*a*c*d*e^38 + 210*sqrt(x*e + d
)*A*a*c*e^39)*e^(-42) - 2/3*(15*(x*e + d)*B*c^2*d^4 - B*c^2*d^5 - 12*(x*e + d)*A*c^2*d^3*e + A*c^2*d^4*e + 18*
(x*e + d)*B*a*c*d^2*e^2 - 2*B*a*c*d^3*e^2 - 12*(x*e + d)*A*a*c*d*e^3 + 2*A*a*c*d^2*e^3 + 3*(x*e + d)*B*a^2*e^4
- B*a^2*d*e^4 + A*a^2*e^5)*e^(-6)/(x*e + d)^(3/2)