3.1240 \(\int \frac{(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=228 \[ -\frac{d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (5 A b e-4 A c d+2 b B d)}{b^3}+\frac{e \sqrt{d+e x} \left (-b c (A e+B d)+2 A c^2 d+3 b^2 B e\right )}{b^2 c^2}+\frac{(d+e x)^{3/2} (b B-2 A c) (c d-b e)}{b^2 c (b+c x)}-\frac{(c d-b e)^{3/2} \left (-b c (2 B d-A e)+4 A c^2 d-3 b^2 B e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{5/2}}-\frac{A (d+e x)^{5/2}}{b x (b+c x)} \]

[Out]

(e*(2*A*c^2*d + 3*b^2*B*e - b*c*(B*d + A*e))*Sqrt[d + e*x])/(b^2*c^2) + ((b*B -
2*A*c)*(c*d - b*e)*(d + e*x)^(3/2))/(b^2*c*(b + c*x)) - (A*(d + e*x)^(5/2))/(b*x
*(b + c*x)) - (d^(3/2)*(2*b*B*d - 4*A*c*d + 5*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[
d]])/b^3 - ((c*d - b*e)^(3/2)*(4*A*c^2*d - 3*b^2*B*e - b*c*(2*B*d - A*e))*ArcTan
h[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*c^(5/2))

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Rubi [A]  time = 1.1487, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (5 A b e-4 A c d+2 b B d)}{b^3}+\frac{e \sqrt{d+e x} \left (-b c (A e+B d)+2 A c^2 d+3 b^2 B e\right )}{b^2 c^2}+\frac{(d+e x)^{3/2} (b B-2 A c) (c d-b e)}{b^2 c (b+c x)}+\frac{(c d-b e)^{3/2} \left (-A b c e-4 A c^2 d+3 b^2 B e+2 b B c d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{5/2}}-\frac{A (d+e x)^{5/2}}{b x (b+c x)} \]

Antiderivative was successfully verified.

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

[Out]

(e*(2*A*c^2*d + 3*b^2*B*e - b*c*(B*d + A*e))*Sqrt[d + e*x])/(b^2*c^2) + ((b*B -
2*A*c)*(c*d - b*e)*(d + e*x)^(3/2))/(b^2*c*(b + c*x)) - (A*(d + e*x)^(5/2))/(b*x
*(b + c*x)) - (d^(3/2)*(2*b*B*d - 4*A*c*d + 5*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[
d]])/b^3 + ((c*d - b*e)^(3/2)*(2*b*B*c*d - 4*A*c^2*d + 3*b^2*B*e - A*b*c*e)*ArcT
anh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*c^(5/2))

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Rubi in Sympy [A]  time = 125.957, size = 211, normalized size = 0.93 \[ \frac{\left (d + e x\right )^{\frac{5}{2}} \left (A c - B b\right )}{b c x \left (b + c x\right )} - \frac{d \left (d + e x\right )^{\frac{3}{2}} \left (2 A c - B b\right )}{b^{2} c x} - \frac{e \sqrt{d + e x} \left (b e \left (A c - 3 B b\right ) - c d \left (2 A c - B b\right )\right )}{b^{2} c^{2}} - \frac{d^{\frac{3}{2}} \left (5 A b e - 4 A c d + 2 B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b^{3}} + \frac{\left (b e - c d\right )^{\frac{3}{2}} \left (A b c e + 4 A c^{2} d - 3 B b^{2} e - 2 B b c d\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b^{3} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d + e*x)**(5/2)*(A*c - B*b)/(b*c*x*(b + c*x)) - d*(d + e*x)**(3/2)*(2*A*c - B*b
)/(b**2*c*x) - e*sqrt(d + e*x)*(b*e*(A*c - 3*B*b) - c*d*(2*A*c - B*b))/(b**2*c**
2) - d**(3/2)*(5*A*b*e - 4*A*c*d + 2*B*b*d)*atanh(sqrt(d + e*x)/sqrt(d))/b**3 +
(b*e - c*d)**(3/2)*(A*b*c*e + 4*A*c**2*d - 3*B*b**2*e - 2*B*b*c*d)*atan(sqrt(c)*
sqrt(d + e*x)/sqrt(b*e - c*d))/(b**3*c**(5/2))

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Mathematica [A]  time = 0.518133, size = 183, normalized size = 0.8 \[ -\frac{d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (5 A b e-4 A c d+2 b B d)}{b^3}+\sqrt{d+e x} \left (\frac{(b B-A c) (c d-b e)^2}{b^2 c^2 (b+c x)}-\frac{A d^2}{b^2 x}+\frac{2 B e^2}{c^2}\right )+\frac{(c d-b e)^{3/2} \left (b c (2 B d-A e)-4 A c^2 d+3 b^2 B e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.034, size = 614, normalized size = 2.7 \[ 2\,{\frac{{e}^{2}B\sqrt{ex+d}}{{c}^{2}}}-{\frac{{e}^{3}A}{c \left ( cex+be \right ) }\sqrt{ex+d}}+2\,{\frac{{e}^{2}\sqrt{ex+d}Ad}{b \left ( cex+be \right ) }}-{\frac{Ac{d}^{2}e}{{b}^{2} \left ( cex+be \right ) }\sqrt{ex+d}}+{\frac{B{e}^{3}b}{{c}^{2} \left ( cex+be \right ) }\sqrt{ex+d}}-2\,{\frac{{e}^{2}B\sqrt{ex+d}d}{c \left ( cex+be \right ) }}+{\frac{Be{d}^{2}}{b \left ( cex+be \right ) }\sqrt{ex+d}}+{\frac{{e}^{3}A}{c}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}+2\,{\frac{{e}^{2}Ad}{b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-7\,{\frac{Ac{d}^{2}e}{{b}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{A{c}^{2}{d}^{3}}{{b}^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-3\,{\frac{B{e}^{3}b}{{c}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{{e}^{2}Bd}{c\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+{\frac{Be{d}^{2}}{b}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}-2\,{\frac{Bc{d}^{3}}{{b}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{{d}^{2}A}{{b}^{2}x}\sqrt{ex+d}}-5\,{\frac{e{d}^{3/2}A}{{b}^{2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }+4\,{\frac{{d}^{5/2}Ac}{{b}^{3}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-2\,{\frac{{d}^{5/2}B}{{b}^{2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*e^2*B*(e*x+d)^(1/2)/c^2-e^3/c*(e*x+d)^(1/2)/(c*e*x+b*e)*A+2*e^2/b*(e*x+d)^(1/2
)/(c*e*x+b*e)*A*d-e/b^2*c*(e*x+d)^(1/2)/(c*e*x+b*e)*A*d^2+e^3*b/c^2*(e*x+d)^(1/2
)/(c*e*x+b*e)*B-2*e^2/c*(e*x+d)^(1/2)/(c*e*x+b*e)*B*d+e/b*(e*x+d)^(1/2)/(c*e*x+b
*e)*B*d^2+e^3/c/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*
A+2*e^2/b/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*d-7*
e/b^2*c/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*d^2+4/
b^3*c^2/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*d^3-3*
e^3*b/c^2/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B+4*e^
2/c/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d+e/b/((b*
e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d^2-2/b^2*c/((b*e-
c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d^3-d^2/b^2*A*(e*x+d
)^(1/2)/x-5*e*d^(3/2)/b^2*arctanh((e*x+d)^(1/2)/d^(1/2))*A+4*d^(5/2)/b^3*arctanh
((e*x+d)^(1/2)/d^(1/2))*A*c-2*d^(5/2)/b^2*arctanh((e*x+d)^(1/2)/d^(1/2))*B

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 10.8542, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(((2*(B*b*c^3 - 2*A*c^4)*d^2 + (B*b^2*c^2 + 3*A*b*c^3)*d*e - (3*B*b^3*c - A
*b^2*c^2)*e^2)*x^2 + (2*(B*b^2*c^2 - 2*A*b*c^3)*d^2 + (B*b^3*c + 3*A*b^2*c^2)*d*
e - (3*B*b^4 - A*b^3*c)*e^2)*x)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2
*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) + ((5*A*b*c^3*d*e + 2*(B*b*c^3
- 2*A*c^4)*d^2)*x^2 + (5*A*b^2*c^2*d*e + 2*(B*b^2*c^2 - 2*A*b*c^3)*d^2)*x)*sqrt(
d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*B*b^3*c*e^2*x^2 - A*b^2*c
^2*d^2 + ((B*b^2*c^2 - 2*A*b*c^3)*d^2 - 2*(B*b^3*c - A*b^2*c^2)*d*e + (3*B*b^4 -
A*b^3*c)*e^2)*x)*sqrt(e*x + d))/(b^3*c^3*x^2 + b^4*c^2*x), 1/2*(2*((2*(B*b*c^3
- 2*A*c^4)*d^2 + (B*b^2*c^2 + 3*A*b*c^3)*d*e - (3*B*b^3*c - A*b^2*c^2)*e^2)*x^2
+ (2*(B*b^2*c^2 - 2*A*b*c^3)*d^2 + (B*b^3*c + 3*A*b^2*c^2)*d*e - (3*B*b^4 - A*b^
3*c)*e^2)*x)*sqrt(-(c*d - b*e)/c)*arctan(sqrt(e*x + d)/sqrt(-(c*d - b*e)/c)) + (
(5*A*b*c^3*d*e + 2*(B*b*c^3 - 2*A*c^4)*d^2)*x^2 + (5*A*b^2*c^2*d*e + 2*(B*b^2*c^
2 - 2*A*b*c^3)*d^2)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*
(2*B*b^3*c*e^2*x^2 - A*b^2*c^2*d^2 + ((B*b^2*c^2 - 2*A*b*c^3)*d^2 - 2*(B*b^3*c -
A*b^2*c^2)*d*e + (3*B*b^4 - A*b^3*c)*e^2)*x)*sqrt(e*x + d))/(b^3*c^3*x^2 + b^4*
c^2*x), -1/2*(2*((5*A*b*c^3*d*e + 2*(B*b*c^3 - 2*A*c^4)*d^2)*x^2 + (5*A*b^2*c^2*
d*e + 2*(B*b^2*c^2 - 2*A*b*c^3)*d^2)*x)*sqrt(-d)*arctan(sqrt(e*x + d)/sqrt(-d))
- ((2*(B*b*c^3 - 2*A*c^4)*d^2 + (B*b^2*c^2 + 3*A*b*c^3)*d*e - (3*B*b^3*c - A*b^2
*c^2)*e^2)*x^2 + (2*(B*b^2*c^2 - 2*A*b*c^3)*d^2 + (B*b^3*c + 3*A*b^2*c^2)*d*e -
(3*B*b^4 - A*b^3*c)*e^2)*x)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqr
t(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) - 2*(2*B*b^3*c*e^2*x^2 - A*b^2*c^2*
d^2 + ((B*b^2*c^2 - 2*A*b*c^3)*d^2 - 2*(B*b^3*c - A*b^2*c^2)*d*e + (3*B*b^4 - A*
b^3*c)*e^2)*x)*sqrt(e*x + d))/(b^3*c^3*x^2 + b^4*c^2*x), -(((5*A*b*c^3*d*e + 2*(
B*b*c^3 - 2*A*c^4)*d^2)*x^2 + (5*A*b^2*c^2*d*e + 2*(B*b^2*c^2 - 2*A*b*c^3)*d^2)*
x)*sqrt(-d)*arctan(sqrt(e*x + d)/sqrt(-d)) - ((2*(B*b*c^3 - 2*A*c^4)*d^2 + (B*b^
2*c^2 + 3*A*b*c^3)*d*e - (3*B*b^3*c - A*b^2*c^2)*e^2)*x^2 + (2*(B*b^2*c^2 - 2*A*
b*c^3)*d^2 + (B*b^3*c + 3*A*b^2*c^2)*d*e - (3*B*b^4 - A*b^3*c)*e^2)*x)*sqrt(-(c*
d - b*e)/c)*arctan(sqrt(e*x + d)/sqrt(-(c*d - b*e)/c)) - (2*B*b^3*c*e^2*x^2 - A*
b^2*c^2*d^2 + ((B*b^2*c^2 - 2*A*b*c^3)*d^2 - 2*(B*b^3*c - A*b^2*c^2)*d*e + (3*B*
b^4 - A*b^3*c)*e^2)*x)*sqrt(e*x + d))/(b^3*c^3*x^2 + b^4*c^2*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.302787, size = 632, normalized size = 2.77 \[ \frac{2 \, \sqrt{x e + d} B e^{2}}{c^{2}} + \frac{{\left (2 \, B b d^{3} - 4 \, A c d^{3} + 5 \, A b d^{2} e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d}} - \frac{{\left (2 \, B b c^{3} d^{3} - 4 \, A c^{4} d^{3} - B b^{2} c^{2} d^{2} e + 7 \, A b c^{3} d^{2} e - 4 \, B b^{3} c d e^{2} - 2 \, A b^{2} c^{2} d e^{2} + 3 \, B b^{4} e^{3} - A b^{3} c e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b^{3} c^{2}} + \frac{{\left (x e + d\right )}^{\frac{3}{2}} B b c^{2} d^{2} e - 2 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{3} d^{2} e - \sqrt{x e + d} B b c^{2} d^{3} e + 2 \, \sqrt{x e + d} A c^{3} d^{3} e - 2 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} c d e^{2} + 2 \,{\left (x e + d\right )}^{\frac{3}{2}} A b c^{2} d e^{2} + 2 \, \sqrt{x e + d} B b^{2} c d^{2} e^{2} - 3 \, \sqrt{x e + d} A b c^{2} d^{2} e^{2} +{\left (x e + d\right )}^{\frac{3}{2}} B b^{3} e^{3} -{\left (x e + d\right )}^{\frac{3}{2}} A b^{2} c e^{3} - \sqrt{x e + d} B b^{3} d e^{3} + \sqrt{x e + d} A b^{2} c d e^{3}}{{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )} b^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x*e + d)*B*e^2/c^2 + (2*B*b*d^3 - 4*A*c*d^3 + 5*A*b*d^2*e)*arctan(sqrt(x*
e + d)/sqrt(-d))/(b^3*sqrt(-d)) - (2*B*b*c^3*d^3 - 4*A*c^4*d^3 - B*b^2*c^2*d^2*e
+ 7*A*b*c^3*d^2*e - 4*B*b^3*c*d*e^2 - 2*A*b^2*c^2*d*e^2 + 3*B*b^4*e^3 - A*b^3*c
*e^3)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e)*b^3*c^2
) + ((x*e + d)^(3/2)*B*b*c^2*d^2*e - 2*(x*e + d)^(3/2)*A*c^3*d^2*e - sqrt(x*e +
d)*B*b*c^2*d^3*e + 2*sqrt(x*e + d)*A*c^3*d^3*e - 2*(x*e + d)^(3/2)*B*b^2*c*d*e^2
+ 2*(x*e + d)^(3/2)*A*b*c^2*d*e^2 + 2*sqrt(x*e + d)*B*b^2*c*d^2*e^2 - 3*sqrt(x*
e + d)*A*b*c^2*d^2*e^2 + (x*e + d)^(3/2)*B*b^3*e^3 - (x*e + d)^(3/2)*A*b^2*c*e^3
- sqrt(x*e + d)*B*b^3*d*e^3 + sqrt(x*e + d)*A*b^2*c*d*e^3)/(((x*e + d)^2*c - 2*
(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)*b^2*c^2)