3.1184 \(\int (A+B x) (d+e x) \left (b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=264 \[ -\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{6144 c^4}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2} \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{384 c^3}-\frac{5 b^6 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{16384 c^{11/2}}+\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{16384 c^5}-\frac{\left (b x+c x^2\right )^{7/2} (-16 c (A e+B d)+9 b B e-14 B c e x)}{112 c^2} \]

[Out]

(5*b^4*(32*A*c^2*d + 9*b^2*B*e - 16*b*c*(B*d + A*e))*(b + 2*c*x)*Sqrt[b*x + c*x^
2])/(16384*c^5) - (5*b^2*(32*A*c^2*d + 9*b^2*B*e - 16*b*c*(B*d + A*e))*(b + 2*c*
x)*(b*x + c*x^2)^(3/2))/(6144*c^4) + ((32*A*c^2*d + 9*b^2*B*e - 16*b*c*(B*d + A*
e))*(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(384*c^3) - ((9*b*B*e - 16*c*(B*d + A*e) -
14*B*c*e*x)*(b*x + c*x^2)^(7/2))/(112*c^2) - (5*b^6*(32*A*c^2*d + 9*b^2*B*e - 16
*b*c*(B*d + A*e))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(16384*c^(11/2))

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Rubi [A]  time = 0.511593, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{6144 c^4}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2} \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{384 c^3}-\frac{5 b^6 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{16384 c^{11/2}}+\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{16384 c^5}-\frac{\left (b x+c x^2\right )^{7/2} (-16 c (A e+B d)+9 b B e-14 B c e x)}{112 c^2} \]

Antiderivative was successfully verified.

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

[Out]

(5*b^4*(32*A*c^2*d + 9*b^2*B*e - 16*b*c*(B*d + A*e))*(b + 2*c*x)*Sqrt[b*x + c*x^
2])/(16384*c^5) - (5*b^2*(32*A*c^2*d + 9*b^2*B*e - 16*b*c*(B*d + A*e))*(b + 2*c*
x)*(b*x + c*x^2)^(3/2))/(6144*c^4) + ((32*A*c^2*d + 9*b^2*B*e - 16*b*c*(B*d + A*
e))*(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(384*c^3) - ((9*b*B*e - 16*c*(B*d + A*e) -
14*B*c*e*x)*(b*x + c*x^2)^(7/2))/(112*c^2) - (5*b^6*(32*A*c^2*d + 9*b^2*B*e - 16
*b*c*(B*d + A*e))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(16384*c^(11/2))

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Rubi in Sympy [A]  time = 29.7417, size = 265, normalized size = 1. \[ \frac{5 b^{6} \left (- 9 B b^{2} e + 16 c \left (- 2 A c d + b \left (A e + B d\right )\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{16384 c^{\frac{11}{2}}} - \frac{5 b^{4} \left (b + 2 c x\right ) \sqrt{b x + c x^{2}} \left (- 9 B b^{2} e + 16 c \left (- 2 A c d + b \left (A e + B d\right )\right )\right )}{16384 c^{5}} + \frac{5 b^{2} \left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}} \left (- 9 B b^{2} e + 16 c \left (- 2 A c d + b \left (A e + B d\right )\right )\right )}{6144 c^{4}} + \frac{\left (b x + c x^{2}\right )^{\frac{7}{2}} \left (- \frac{9 B b e}{2} + 7 B c e x + 8 c \left (A e + B d\right )\right )}{56 c^{2}} - \frac{\left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}} \left (- 9 B b^{2} e + 16 c \left (- 2 A c d + b \left (A e + B d\right )\right )\right )}{384 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)*(c*x**2+b*x)**(5/2),x)

[Out]

5*b**6*(-9*B*b**2*e + 16*c*(-2*A*c*d + b*(A*e + B*d)))*atanh(sqrt(c)*x/sqrt(b*x
+ c*x**2))/(16384*c**(11/2)) - 5*b**4*(b + 2*c*x)*sqrt(b*x + c*x**2)*(-9*B*b**2*
e + 16*c*(-2*A*c*d + b*(A*e + B*d)))/(16384*c**5) + 5*b**2*(b + 2*c*x)*(b*x + c*
x**2)**(3/2)*(-9*B*b**2*e + 16*c*(-2*A*c*d + b*(A*e + B*d)))/(6144*c**4) + (b*x
+ c*x**2)**(7/2)*(-9*B*b*e/2 + 7*B*c*e*x + 8*c*(A*e + B*d))/(56*c**2) - (b + 2*c
*x)*(b*x + c*x**2)**(5/2)*(-9*B*b**2*e + 16*c*(-2*A*c*d + b*(A*e + B*d)))/(384*c
**3)

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Mathematica [A]  time = 0.827611, size = 330, normalized size = 1.25 \[ \frac{(x (b+c x))^{5/2} \left (\frac{\sqrt{x} \left (-210 b^6 c (8 A e+8 B d+3 B e x)+56 b^5 c^2 (20 A (3 d+e x)+B x (20 d+9 e x))-16 b^4 c^3 x (28 A (5 d+2 e x)+B x (56 d+27 e x))+128 b^3 c^4 x^2 (2 A (7 d+3 e x)+3 B x (2 d+e x))+256 b^2 c^5 x^3 (A (378 d+296 e x)+B x (296 d+243 e x))+1024 b c^6 x^4 (4 A (35 d+29 e x)+B x (116 d+99 e x))+2048 c^7 x^5 (4 A (7 d+6 e x)+3 B x (8 d+7 e x))+945 b^7 B e\right )}{21 c^5 (b+c x)^2}-\frac{5 b^6 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right ) \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{c^{11/2} (b+c x)^{5/2}}\right )}{16384 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((x*(b + c*x))^(5/2)*((Sqrt[x]*(945*b^7*B*e - 210*b^6*c*(8*B*d + 8*A*e + 3*B*e*x
) + 128*b^3*c^4*x^2*(3*B*x*(2*d + e*x) + 2*A*(7*d + 3*e*x)) + 2048*c^7*x^5*(4*A*
(7*d + 6*e*x) + 3*B*x*(8*d + 7*e*x)) + 56*b^5*c^2*(20*A*(3*d + e*x) + B*x*(20*d
+ 9*e*x)) - 16*b^4*c^3*x*(28*A*(5*d + 2*e*x) + B*x*(56*d + 27*e*x)) + 1024*b*c^6
*x^4*(4*A*(35*d + 29*e*x) + B*x*(116*d + 99*e*x)) + 256*b^2*c^5*x^3*(B*x*(296*d
+ 243*e*x) + A*(378*d + 296*e*x))))/(21*c^5*(b + c*x)^2) - (5*b^6*(32*A*c^2*d +
9*b^2*B*e - 16*b*c*(B*d + A*e))*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(c^(11/2
)*(b + c*x)^(5/2))))/(16384*x^(5/2))

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Maple [B]  time = 0.013, size = 716, normalized size = 2.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)*(c*x^2+b*x)^(5/2),x)

[Out]

1/6*d*A*(c*x^2+b*x)^(5/2)*x+1/7*(c*x^2+b*x)^(7/2)/c*A*e+1/7*(c*x^2+b*x)^(7/2)/c*
B*d+1/12*d*A/c*(c*x^2+b*x)^(5/2)*b+5/2048*b^7/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*
x^2+b*x)^(1/2))*A*e-9/112*B*e*b/c^2*(c*x^2+b*x)^(7/2)+5/512*d*A*b^5/c^3*(c*x^2+b
*x)^(1/2)+3/128*B*e*b^3/c^3*(c*x^2+b*x)^(5/2)-15/2048*B*e*b^5/c^4*(c*x^2+b*x)^(3
/2)+45/16384*B*e*b^7/c^5*(c*x^2+b*x)^(1/2)-45/32768*B*e*b^8/c^(11/2)*ln((1/2*b+c
*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+5/2048*b^7/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+
b*x)^(1/2))*B*d-5/192*d*A*b^3/c^2*(c*x^2+b*x)^(3/2)-1/24*b^2/c^2*(c*x^2+b*x)^(5/
2)*A*e-1/24*b^2/c^2*(c*x^2+b*x)^(5/2)*B*d+5/384*b^4/c^3*(c*x^2+b*x)^(3/2)*A*e-5/
1024*d*A*b^6/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+5/384*b^4/c^3*(c*
x^2+b*x)^(3/2)*B*d-5/1024*b^6/c^4*(c*x^2+b*x)^(1/2)*A*e-5/1024*b^6/c^4*(c*x^2+b*
x)^(1/2)*B*d+1/8*B*e*x*(c*x^2+b*x)^(7/2)/c+3/64*B*e*b^2/c^2*(c*x^2+b*x)^(5/2)*x-
15/1024*B*e*b^4/c^3*(c*x^2+b*x)^(3/2)*x+45/8192*B*e*b^6/c^4*(c*x^2+b*x)^(1/2)*x+
5/256*d*A*b^4/c^2*(c*x^2+b*x)^(1/2)*x+5/192*b^3/c^2*(c*x^2+b*x)^(3/2)*x*B*d-5/51
2*b^5/c^3*(c*x^2+b*x)^(1/2)*x*A*e-1/12*b/c*(c*x^2+b*x)^(5/2)*x*B*d+5/192*b^3/c^2
*(c*x^2+b*x)^(3/2)*x*A*e-5/96*d*A*b^2/c*(c*x^2+b*x)^(3/2)*x-5/512*b^5/c^3*(c*x^2
+b*x)^(1/2)*x*B*d-1/12*b/c*(c*x^2+b*x)^(5/2)*x*A*e

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)*(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.30333, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)*(e*x + d),x, algorithm="fricas")

[Out]

[1/688128*(2*(43008*B*c^7*e*x^7 + 3072*(16*B*c^7*d + (33*B*b*c^6 + 16*A*c^7)*e)*
x^6 + 256*(16*(29*B*b*c^6 + 14*A*c^7)*d + (243*B*b^2*c^5 + 464*A*b*c^6)*e)*x^5 +
128*(16*(37*B*b^2*c^5 + 70*A*b*c^6)*d + (3*B*b^3*c^4 + 592*A*b^2*c^5)*e)*x^4 +
48*(16*(B*b^3*c^4 + 126*A*b^2*c^5)*d - (9*B*b^4*c^3 - 16*A*b^3*c^4)*e)*x^3 - 56*
(16*(B*b^4*c^3 - 2*A*b^3*c^4)*d - (9*B*b^5*c^2 - 16*A*b^4*c^3)*e)*x^2 - 1680*(B*
b^6*c - 2*A*b^5*c^2)*d + 105*(9*B*b^7 - 16*A*b^6*c)*e + 70*(16*(B*b^5*c^2 - 2*A*
b^4*c^3)*d - (9*B*b^6*c - 16*A*b^5*c^2)*e)*x)*sqrt(c*x^2 + b*x)*sqrt(c) + 105*(1
6*(B*b^7*c - 2*A*b^6*c^2)*d - (9*B*b^8 - 16*A*b^7*c)*e)*log((2*c*x + b)*sqrt(c)
+ 2*sqrt(c*x^2 + b*x)*c))/c^(11/2), 1/344064*((43008*B*c^7*e*x^7 + 3072*(16*B*c^
7*d + (33*B*b*c^6 + 16*A*c^7)*e)*x^6 + 256*(16*(29*B*b*c^6 + 14*A*c^7)*d + (243*
B*b^2*c^5 + 464*A*b*c^6)*e)*x^5 + 128*(16*(37*B*b^2*c^5 + 70*A*b*c^6)*d + (3*B*b
^3*c^4 + 592*A*b^2*c^5)*e)*x^4 + 48*(16*(B*b^3*c^4 + 126*A*b^2*c^5)*d - (9*B*b^4
*c^3 - 16*A*b^3*c^4)*e)*x^3 - 56*(16*(B*b^4*c^3 - 2*A*b^3*c^4)*d - (9*B*b^5*c^2
- 16*A*b^4*c^3)*e)*x^2 - 1680*(B*b^6*c - 2*A*b^5*c^2)*d + 105*(9*B*b^7 - 16*A*b^
6*c)*e + 70*(16*(B*b^5*c^2 - 2*A*b^4*c^3)*d - (9*B*b^6*c - 16*A*b^5*c^2)*e)*x)*s
qrt(c*x^2 + b*x)*sqrt(-c) + 105*(16*(B*b^7*c - 2*A*b^6*c^2)*d - (9*B*b^8 - 16*A*
b^7*c)*e)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/(sqrt(-c)*c^5)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (A + B x\right ) \left (d + e x\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)*(c*x**2+b*x)**(5/2),x)

[Out]

Integral((x*(b + c*x))**(5/2)*(A + B*x)*(d + e*x), x)

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GIAC/XCAS [A]  time = 0.287168, size = 574, normalized size = 2.17 \[ \frac{1}{344064} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (12 \,{\left (14 \, B c^{2} x e + \frac{16 \, B c^{9} d + 33 \, B b c^{8} e + 16 \, A c^{9} e}{c^{7}}\right )} x + \frac{464 \, B b c^{8} d + 224 \, A c^{9} d + 243 \, B b^{2} c^{7} e + 464 \, A b c^{8} e}{c^{7}}\right )} x + \frac{592 \, B b^{2} c^{7} d + 1120 \, A b c^{8} d + 3 \, B b^{3} c^{6} e + 592 \, A b^{2} c^{7} e}{c^{7}}\right )} x + \frac{3 \,{\left (16 \, B b^{3} c^{6} d + 2016 \, A b^{2} c^{7} d - 9 \, B b^{4} c^{5} e + 16 \, A b^{3} c^{6} e\right )}}{c^{7}}\right )} x - \frac{7 \,{\left (16 \, B b^{4} c^{5} d - 32 \, A b^{3} c^{6} d - 9 \, B b^{5} c^{4} e + 16 \, A b^{4} c^{5} e\right )}}{c^{7}}\right )} x + \frac{35 \,{\left (16 \, B b^{5} c^{4} d - 32 \, A b^{4} c^{5} d - 9 \, B b^{6} c^{3} e + 16 \, A b^{5} c^{4} e\right )}}{c^{7}}\right )} x - \frac{105 \,{\left (16 \, B b^{6} c^{3} d - 32 \, A b^{5} c^{4} d - 9 \, B b^{7} c^{2} e + 16 \, A b^{6} c^{3} e\right )}}{c^{7}}\right )} - \frac{5 \,{\left (16 \, B b^{7} c d - 32 \, A b^{6} c^{2} d - 9 \, B b^{8} e + 16 \, A b^{7} c e\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{32768 \, c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)*(e*x + d),x, algorithm="giac")

[Out]

1/344064*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(2*(12*(14*B*c^2*x*e + (16*B*c^9*d + 33*B
*b*c^8*e + 16*A*c^9*e)/c^7)*x + (464*B*b*c^8*d + 224*A*c^9*d + 243*B*b^2*c^7*e +
464*A*b*c^8*e)/c^7)*x + (592*B*b^2*c^7*d + 1120*A*b*c^8*d + 3*B*b^3*c^6*e + 592
*A*b^2*c^7*e)/c^7)*x + 3*(16*B*b^3*c^6*d + 2016*A*b^2*c^7*d - 9*B*b^4*c^5*e + 16
*A*b^3*c^6*e)/c^7)*x - 7*(16*B*b^4*c^5*d - 32*A*b^3*c^6*d - 9*B*b^5*c^4*e + 16*A
*b^4*c^5*e)/c^7)*x + 35*(16*B*b^5*c^4*d - 32*A*b^4*c^5*d - 9*B*b^6*c^3*e + 16*A*
b^5*c^4*e)/c^7)*x - 105*(16*B*b^6*c^3*d - 32*A*b^5*c^4*d - 9*B*b^7*c^2*e + 16*A*
b^6*c^3*e)/c^7) - 5/32768*(16*B*b^7*c*d - 32*A*b^6*c^2*d - 9*B*b^8*e + 16*A*b^7*
c*e)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(11/2)