3.1440 \(\int \frac{1}{(a+b x) (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=93 \[ -\frac{2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{5/2}}+\frac{2 b}{\sqrt{c+d x} (b c-a d)^2}+\frac{2}{3 (c+d x)^{3/2} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.114274, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{5/2}}+\frac{2 b}{\sqrt{c+d x} (b c-a d)^2}+\frac{2}{3 (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)*(c + d*x)^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 18.9103, size = 80, normalized size = 0.86 \[ \frac{2 b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{\left (a d - b c\right )^{\frac{5}{2}}} + \frac{2 b}{\sqrt{c + d x} \left (a d - b c\right )^{2}} - \frac{2}{3 \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)/(d*x+c)**(5/2),x)

[Out]

2*b**(3/2)*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c))/(a*d - b*c)**(5/2) + 2*b/
(sqrt(c + d*x)*(a*d - b*c)**2) - 2/(3*(c + d*x)**(3/2)*(a*d - b*c))

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Mathematica [A]  time = 0.309396, size = 85, normalized size = 0.91 \[ \frac{2 (-a d+4 b c+3 b d x)}{3 (c+d x)^{3/2} (b c-a d)^2}-\frac{2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)*(c + d*x)^(5/2)),x]

[Out]

(2*(4*b*c - a*d + 3*b*d*x))/(3*(b*c - a*d)^2*(c + d*x)^(3/2)) - (2*b^(3/2)*ArcTa
nh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b*c - a*d)^(5/2)

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Maple [A]  time = 0.023, size = 90, normalized size = 1. \[ -{\frac{2}{3\,ad-3\,bc} \left ( dx+c \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{b}{ \left ( ad-bc \right ) ^{2}\sqrt{dx+c}}}+2\,{\frac{{b}^{2}}{ \left ( ad-bc \right ) ^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)/(d*x+c)^(5/2),x)

[Out]

-2/3/(a*d-b*c)/(d*x+c)^(3/2)+2*b/(a*d-b*c)^2/(d*x+c)^(1/2)+2*b^2/(a*d-b*c)^2/((a
*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.21973, size = 1, normalized size = 0.01 \[ \left [\frac{6 \, b d x + 3 \,{\left (b d x + b c\right )} \sqrt{d x + c} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x + 2 \, b c - a d - 2 \,{\left (b c - a d\right )} \sqrt{d x + c} \sqrt{\frac{b}{b c - a d}}}{b x + a}\right ) + 8 \, b c - 2 \, a d}{3 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} \sqrt{d x + c}}, \frac{2 \,{\left (3 \, b d x - 3 \,{\left (b d x + b c\right )} \sqrt{d x + c} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}{\sqrt{d x + c} b}\right ) + 4 \, b c - a d\right )}}{3 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} \sqrt{d x + c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*(d*x + c)^(5/2)),x, algorithm="fricas")

[Out]

[1/3*(6*b*d*x + 3*(b*d*x + b*c)*sqrt(d*x + c)*sqrt(b/(b*c - a*d))*log((b*d*x + 2
*b*c - a*d - 2*(b*c - a*d)*sqrt(d*x + c)*sqrt(b/(b*c - a*d)))/(b*x + a)) + 8*b*c
- 2*a*d)/((b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2 + (b^2*c^2*d - 2*a*b*c*d^2 + a^2*d
^3)*x)*sqrt(d*x + c)), 2/3*(3*b*d*x - 3*(b*d*x + b*c)*sqrt(d*x + c)*sqrt(-b/(b*c
- a*d))*arctan(-(b*c - a*d)*sqrt(-b/(b*c - a*d))/(sqrt(d*x + c)*b)) + 4*b*c - a
*d)/((b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2 + (b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*x)
*sqrt(d*x + c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right ) \left (c + d x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)/(d*x+c)**(5/2),x)

[Out]

Integral(1/((a + b*x)*(c + d*x)**(5/2)), x)

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GIAC/XCAS [A]  time = 0.221123, size = 153, normalized size = 1.65 \[ \frac{2 \, b^{2} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-b^{2} c + a b d}} + \frac{2 \,{\left (3 \,{\left (d x + c\right )} b + b c - a d\right )}}{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\left (d x + c\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*(d*x + c)^(5/2)),x, algorithm="giac")

[Out]

2*b^2*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^2*c^2 - 2*a*b*c*d + a^2*d
^2)*sqrt(-b^2*c + a*b*d)) + 2/3*(3*(d*x + c)*b + b*c - a*d)/((b^2*c^2 - 2*a*b*c*
d + a^2*d^2)*(d*x + c)^(3/2))