3.1441 \(\int \frac{1}{(a+b x)^2 (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=124 \[ \frac{5 b^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{7/2}}-\frac{5 b d}{\sqrt{c+d x} (b c-a d)^3}-\frac{1}{(a+b x) (c+d x)^{3/2} (b c-a d)}-\frac{5 d}{3 (c+d x)^{3/2} (b c-a d)^2} \]

[Out]

(-5*d)/(3*(b*c - a*d)^2*(c + d*x)^(3/2)) - 1/((b*c - a*d)*(a + b*x)*(c + d*x)^(3
/2)) - (5*b*d)/((b*c - a*d)^3*Sqrt[c + d*x]) + (5*b^(3/2)*d*ArcTanh[(Sqrt[b]*Sqr
t[c + d*x])/Sqrt[b*c - a*d]])/(b*c - a*d)^(7/2)

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Rubi [A]  time = 0.149603, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{5 b^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{7/2}}-\frac{5 b d}{\sqrt{c+d x} (b c-a d)^3}-\frac{1}{(a+b x) (c+d x)^{3/2} (b c-a d)}-\frac{5 d}{3 (c+d x)^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

(-5*d)/(3*(b*c - a*d)^2*(c + d*x)^(3/2)) - 1/((b*c - a*d)*(a + b*x)*(c + d*x)^(3
/2)) - (5*b*d)/((b*c - a*d)^3*Sqrt[c + d*x]) + (5*b^(3/2)*d*ArcTanh[(Sqrt[b]*Sqr
t[c + d*x])/Sqrt[b*c - a*d]])/(b*c - a*d)^(7/2)

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Rubi in Sympy [A]  time = 27.5669, size = 109, normalized size = 0.88 \[ \frac{5 b^{\frac{3}{2}} d \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{\left (a d - b c\right )^{\frac{7}{2}}} + \frac{5 b d}{\sqrt{c + d x} \left (a d - b c\right )^{3}} - \frac{5 d}{3 \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} + \frac{1}{\left (a + b x\right ) \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*b**(3/2)*d*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c))/(a*d - b*c)**(7/2) + 5*
b*d/(sqrt(c + d*x)*(a*d - b*c)**3) - 5*d/(3*(c + d*x)**(3/2)*(a*d - b*c)**2) + 1
/((a + b*x)*(c + d*x)**(3/2)*(a*d - b*c))

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Mathematica [A]  time = 0.24182, size = 125, normalized size = 1.01 \[ \frac{5 b^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{7/2}}+\sqrt{c+d x} \left (-\frac{b^2}{(a+b x) (b c-a d)^3}-\frac{4 b d}{(c+d x) (b c-a d)^3}-\frac{2 d}{3 (c+d x)^2 (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.027, size = 125, normalized size = 1. \[ -{\frac{2\,d}{3\, \left ( ad-bc \right ) ^{2}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}}+4\,{\frac{bd}{ \left ( ad-bc \right ) ^{3}\sqrt{dx+c}}}+{\frac{{b}^{2}d}{ \left ( ad-bc \right ) ^{3} \left ( bdx+ad \right ) }\sqrt{dx+c}}+5\,{\frac{{b}^{2}d}{ \left ( ad-bc \right ) ^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*d/(a*d-b*c)^2/(d*x+c)^(3/2)+4*d/(a*d-b*c)^3*b/(d*x+c)^(1/2)+d*b^2/(a*d-b*c)
^3*(d*x+c)^(1/2)/(b*d*x+a*d)+5*d*b^2/(a*d-b*c)^3/((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)^2*(d*x + c)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227061, size = 1, normalized size = 0.01 \[ \left [-\frac{30 \, b^{2} d^{2} x^{2} + 6 \, b^{2} c^{2} + 28 \, a b c d - 4 \, a^{2} d^{2} + 15 \,{\left (b^{2} d^{2} x^{2} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{d x + c} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x + 2 \, b c - a d - 2 \,{\left (b c - a d\right )} \sqrt{d x + c} \sqrt{\frac{b}{b c - a d}}}{b x + a}\right ) + 20 \,{\left (2 \, b^{2} c d + a b d^{2}\right )} x}{6 \,{\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} +{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x\right )} \sqrt{d x + c}}, -\frac{15 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 14 \, a b c d - 2 \, a^{2} d^{2} - 15 \,{\left (b^{2} d^{2} x^{2} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{d x + c} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}{\sqrt{d x + c} b}\right ) + 10 \,{\left (2 \, b^{2} c d + a b d^{2}\right )} x}{3 \,{\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} +{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x\right )} \sqrt{d x + c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*(30*b^2*d^2*x^2 + 6*b^2*c^2 + 28*a*b*c*d - 4*a^2*d^2 + 15*(b^2*d^2*x^2 + a
*b*c*d + (b^2*c*d + a*b*d^2)*x)*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)) + 20*(2
*b^2*c*d + a*b*d^2)*x)/((a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d
^3 + (b^4*c^3*d - 3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*x^2 + (b^4*c^4
- 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*x)*sqrt(d*x + c)), -1/3*(15*b^2*d^2*x
^2 + 3*b^2*c^2 + 14*a*b*c*d - 2*a^2*d^2 - 15*(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d +
a*b*d^2)*x)*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)) + 10*(2*b^2*c*d + a*b*d^2)*x)/((a*b^3*c^4 - 3*a^2*b^
2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3 + (b^4*c^3*d - 3*a*b^3*c^2*d^2 + 3*a^2*b^2
*c*d^3 - a^3*b*d^4)*x^2 + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*x)
*sqrt(d*x + c))]

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Sympy [F(-2)]  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)**2/(d*x+c)**(5/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.220147, size = 292, normalized size = 2.35 \[ -\frac{5 \, b^{2} d \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{-b^{2} c + a b d}} - \frac{\sqrt{d x + c} b^{2} d}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}} - \frac{2 \,{\left (6 \,{\left (d x + c\right )} b d + b c d - a d^{2}\right )}}{3 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left (d x + c\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*b^2*d*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^3*c^3 - 3*a*b^2*c^2*d
+ 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(-b^2*c + a*b*d)) - sqrt(d*x + c)*b^2*d/((b^3*c^3
- 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*((d*x + c)*b - b*c + a*d)) - 2/3*(6*
(d*x + c)*b*d + b*c*d - a*d^2)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d
^3)*(d*x + c)^(3/2))