3.1235 \(\int \frac{A+B x}{(d+e x)^{5/2} \left (b x+c x^2\right )} \, dx\)

Optimal. Leaf size=164 \[ -\frac{2 c^{3/2} (b B-A c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b (c d-b e)^{5/2}}+\frac{2 \left (B c d^2-A e (2 c d-b e)\right )}{d^2 \sqrt{d+e x} (c d-b e)^2}+\frac{2 (B d-A e)}{3 d (d+e x)^{3/2} (c d-b e)}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b d^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.769799, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{2 c^{3/2} (b B-A c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b (c d-b e)^{5/2}}+\frac{2 \left (B c d^2-A e (2 c d-b e)\right )}{d^2 \sqrt{d+e x} (c d-b e)^2}+\frac{2 (B d-A e)}{3 d (d+e x)^{3/2} (c d-b e)}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b d^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 74.4536, size = 150, normalized size = 0.91 \[ - \frac{2 A \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b d^{\frac{5}{2}}} + \frac{2 \left (A e - B d\right )}{3 d \left (d + e x\right )^{\frac{3}{2}} \left (b e - c d\right )} + \frac{2 \left (A b e^{2} - 2 A c d e + B c d^{2}\right )}{d^{2} \sqrt{d + e x} \left (b e - c d\right )^{2}} - \frac{2 c^{\frac{3}{2}} \left (A c - B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b \left (b e - c d\right )^{\frac{5}{2}}} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.933877, size = 159, normalized size = 0.97 \[ \frac{2 c^{3/2} (A c-b B) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b (c d-b e)^{5/2}}+\frac{2 \left (A e (b e (4 d+3 e x)-c d (7 d+6 e x))+B d^2 (-b e+4 c d+3 c e x)\right )}{3 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b d^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.028, size = 243, normalized size = 1.5 \[{\frac{2\,Ae}{3\,d \left ( be-cd \right ) } \left ( ex+d \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,B}{3\,be-3\,cd} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{Ab{e}^{2}}{{d}^{2} \left ( be-cd \right ) ^{2}\sqrt{ex+d}}}-4\,{\frac{Ace}{d \left ( be-cd \right ) ^{2}\sqrt{ex+d}}}+2\,{\frac{Bc}{ \left ( be-cd \right ) ^{2}\sqrt{ex+d}}}-2\,{\frac{A{c}^{3}}{ \left ( be-cd \right ) ^{2}b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+2\,{\frac{B{c}^{2}}{ \left ( be-cd \right ) ^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{A}{b{d}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

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),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.81476, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/3*(3*((B*b*c - A*c^2)*d^2*e*x + (B*b*c - A*c^2)*d^3)*sqrt(e*x + d)*sqrt(d)*s
qrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c
/(c*d - b*e)))/(c*x + b)) - 3*(A*c^2*d^3 - 2*A*b*c*d^2*e + A*b^2*d*e^2 + (A*c^2*
d^2*e - 2*A*b*c*d*e^2 + A*b^2*e^3)*x)*sqrt(e*x + d)*log(((e*x + 2*d)*sqrt(d) - 2
*sqrt(e*x + d)*d)/x) - 2*(4*B*b*c*d^3 + 4*A*b^2*d*e^2 - (B*b^2 + 7*A*b*c)*d^2*e
+ 3*(B*b*c*d^2*e - 2*A*b*c*d*e^2 + A*b^2*e^3)*x)*sqrt(d))/((b*c^2*d^5 - 2*b^2*c*
d^4*e + b^3*d^3*e^2 + (b*c^2*d^4*e - 2*b^2*c*d^3*e^2 + b^3*d^2*e^3)*x)*sqrt(e*x
+ d)*sqrt(d)), -1/3*(6*((B*b*c - A*c^2)*d^2*e*x + (B*b*c - A*c^2)*d^3)*sqrt(e*x
+ d)*sqrt(d)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(-c/(c*d - b*e))/(sqrt
(e*x + d)*c)) - 3*(A*c^2*d^3 - 2*A*b*c*d^2*e + A*b^2*d*e^2 + (A*c^2*d^2*e - 2*A*
b*c*d*e^2 + A*b^2*e^3)*x)*sqrt(e*x + d)*log(((e*x + 2*d)*sqrt(d) - 2*sqrt(e*x +
d)*d)/x) - 2*(4*B*b*c*d^3 + 4*A*b^2*d*e^2 - (B*b^2 + 7*A*b*c)*d^2*e + 3*(B*b*c*d
^2*e - 2*A*b*c*d*e^2 + A*b^2*e^3)*x)*sqrt(d))/((b*c^2*d^5 - 2*b^2*c*d^4*e + b^3*
d^3*e^2 + (b*c^2*d^4*e - 2*b^2*c*d^3*e^2 + b^3*d^2*e^3)*x)*sqrt(e*x + d)*sqrt(d)
), -1/3*(3*((B*b*c - A*c^2)*d^2*e*x + (B*b*c - A*c^2)*d^3)*sqrt(e*x + d)*sqrt(-d
)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqr
t(c/(c*d - b*e)))/(c*x + b)) - 6*(A*c^2*d^3 - 2*A*b*c*d^2*e + A*b^2*d*e^2 + (A*c
^2*d^2*e - 2*A*b*c*d*e^2 + A*b^2*e^3)*x)*sqrt(e*x + d)*arctan(d/(sqrt(e*x + d)*s
qrt(-d))) - 2*(4*B*b*c*d^3 + 4*A*b^2*d*e^2 - (B*b^2 + 7*A*b*c)*d^2*e + 3*(B*b*c*
d^2*e - 2*A*b*c*d*e^2 + A*b^2*e^3)*x)*sqrt(-d))/((b*c^2*d^5 - 2*b^2*c*d^4*e + b^
3*d^3*e^2 + (b*c^2*d^4*e - 2*b^2*c*d^3*e^2 + b^3*d^2*e^3)*x)*sqrt(e*x + d)*sqrt(
-d)), -2/3*(3*((B*b*c - A*c^2)*d^2*e*x + (B*b*c - A*c^2)*d^3)*sqrt(e*x + d)*sqrt
(-d)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(-c/(c*d - b*e))/(sqrt(e*x + d
)*c)) - 3*(A*c^2*d^3 - 2*A*b*c*d^2*e + A*b^2*d*e^2 + (A*c^2*d^2*e - 2*A*b*c*d*e^
2 + A*b^2*e^3)*x)*sqrt(e*x + d)*arctan(d/(sqrt(e*x + d)*sqrt(-d))) - (4*B*b*c*d^
3 + 4*A*b^2*d*e^2 - (B*b^2 + 7*A*b*c)*d^2*e + 3*(B*b*c*d^2*e - 2*A*b*c*d*e^2 + A
*b^2*e^3)*x)*sqrt(-d))/((b*c^2*d^5 - 2*b^2*c*d^4*e + b^3*d^3*e^2 + (b*c^2*d^4*e
- 2*b^2*c*d^3*e^2 + b^3*d^2*e^3)*x)*sqrt(e*x + d)*sqrt(-d))]

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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),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.28463, size = 293, normalized size = 1.79 \[ \frac{2 \,{\left (B b c^{2} - A c^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{{\left (b c^{2} d^{2} - 2 \, b^{2} c d e + b^{3} e^{2}\right )} \sqrt{-c^{2} d + b c e}} + \frac{2 \,{\left (3 \,{\left (x e + d\right )} B c d^{2} + B c d^{3} - 6 \,{\left (x e + d\right )} A c d e - B b d^{2} e - A c d^{2} e + 3 \,{\left (x e + d\right )} A b e^{2} + A b d e^{2}\right )}}{3 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )}{\left (x e + d\right )}^{\frac{3}{2}}} + \frac{2 \, A \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b \sqrt{-d} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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