3.672 \(\int \frac{(a+b x)^{5/2}}{(c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=138 \[ \frac{15 \sqrt{b} (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 d^{7/2}}-\frac{15 b \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}{4 d^3}+\frac{5 b (a+b x)^{3/2} \sqrt{c+d x}}{2 d^2}-\frac{2 (a+b x)^{5/2}}{d \sqrt{c+d x}} \]

[Out]

(-2*(a + b*x)^(5/2))/(d*Sqrt[c + d*x]) - (15*b*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c
+ d*x])/(4*d^3) + (5*b*(a + b*x)^(3/2)*Sqrt[c + d*x])/(2*d^2) + (15*Sqrt[b]*(b*c
- a*d)^2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*d^(7/2))

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Rubi [A]  time = 0.170735, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{15 \sqrt{b} (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 d^{7/2}}-\frac{15 b \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}{4 d^3}+\frac{5 b (a+b x)^{3/2} \sqrt{c+d x}}{2 d^2}-\frac{2 (a+b x)^{5/2}}{d \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(5/2)/(c + d*x)^(3/2),x]

[Out]

(-2*(a + b*x)^(5/2))/(d*Sqrt[c + d*x]) - (15*b*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c
+ d*x])/(4*d^3) + (5*b*(a + b*x)^(3/2)*Sqrt[c + d*x])/(2*d^2) + (15*Sqrt[b]*(b*c
- a*d)^2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*d^(7/2))

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Rubi in Sympy [A]  time = 20.957, size = 128, normalized size = 0.93 \[ \frac{15 \sqrt{b} \left (a d - b c\right )^{2} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{4 d^{\frac{7}{2}}} + \frac{5 b \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x}}{2 d^{2}} + \frac{15 b \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )}{4 d^{3}} - \frac{2 \left (a + b x\right )^{\frac{5}{2}}}{d \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)/(d*x+c)**(3/2),x)

[Out]

15*sqrt(b)*(a*d - b*c)**2*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(
4*d**(7/2)) + 5*b*(a + b*x)**(3/2)*sqrt(c + d*x)/(2*d**2) + 15*b*sqrt(a + b*x)*s
qrt(c + d*x)*(a*d - b*c)/(4*d**3) - 2*(a + b*x)**(5/2)/(d*sqrt(c + d*x))

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Mathematica [A]  time = 0.164366, size = 138, normalized size = 1. \[ \sqrt{a+b x} \sqrt{c+d x} \left (-\frac{2 (a d-b c)^2}{d^3 (c+d x)}-\frac{b (7 b c-9 a d)}{4 d^3}+\frac{b^2 x}{2 d^2}\right )+\frac{15 \sqrt{b} (b c-a d)^2 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{8 d^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(5/2)/(c + d*x)^(3/2),x]

[Out]

Sqrt[a + b*x]*Sqrt[c + d*x]*(-(b*(7*b*c - 9*a*d))/(4*d^3) + (b^2*x)/(2*d^2) - (2
*(-(b*c) + a*d)^2)/(d^3*(c + d*x))) + (15*Sqrt[b]*(b*c - a*d)^2*Log[b*c + a*d +
2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(8*d^(7/2))

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Maple [F]  time = 0., size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{5}{2}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)/(d*x+c)^(3/2),x)

[Out]

int((b*x+a)^(5/2)/(d*x+c)^(3/2),x)

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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)^(5/2)/(d*x + c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.487824, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} \sqrt{\frac{b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{b}{d}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \,{\left (2 \, b^{2} d^{2} x^{2} - 15 \, b^{2} c^{2} + 25 \, a b c d - 8 \, a^{2} d^{2} -{\left (5 \, b^{2} c d - 9 \, a b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{16 \,{\left (d^{4} x + c d^{3}\right )}}, \frac{15 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} \sqrt{-\frac{b}{d}} \arctan \left (\frac{2 \, b d x + b c + a d}{2 \, \sqrt{b x + a} \sqrt{d x + c} d \sqrt{-\frac{b}{d}}}\right ) + 2 \,{\left (2 \, b^{2} d^{2} x^{2} - 15 \, b^{2} c^{2} + 25 \, a b c d - 8 \, a^{2} d^{2} -{\left (5 \, b^{2} c d - 9 \, a b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{8 \,{\left (d^{4} x + c d^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)/(d*x + c)^(3/2),x, algorithm="fricas")

[Out]

[1/16*(15*(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(b/d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d^2*x
+ b*c*d + a*d^2)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b/d) + 8*(b^2*c*d + a*b*d^2)*
x) + 4*(2*b^2*d^2*x^2 - 15*b^2*c^2 + 25*a*b*c*d - 8*a^2*d^2 - (5*b^2*c*d - 9*a*b
*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(d^4*x + c*d^3), 1/8*(15*(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(-b/d)*arctan(1/2
*(2*b*d*x + b*c + a*d)/(sqrt(b*x + a)*sqrt(d*x + c)*d*sqrt(-b/d))) + 2*(2*b^2*d^
2*x^2 - 15*b^2*c^2 + 25*a*b*c*d - 8*a^2*d^2 - (5*b^2*c*d - 9*a*b*d^2)*x)*sqrt(b*
x + a)*sqrt(d*x + c))/(d^4*x + c*d^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{\frac{5}{2}}}{\left (c + d x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(5/2)/(d*x+c)**(3/2),x)

[Out]

Integral((a + b*x)**(5/2)/(c + d*x)**(3/2), x)

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GIAC/XCAS [A]  time = 0.241255, size = 302, normalized size = 2.19 \[ \frac{{\left ({\left (\frac{2 \,{\left (b x + a\right )} b^{2} d^{4}}{b^{8} c d^{6} - a b^{7} d^{7}} - \frac{5 \,{\left (b^{3} c d^{3} - a b^{2} d^{4}\right )}}{b^{8} c d^{6} - a b^{7} d^{7}}\right )}{\left (b x + a\right )} - \frac{15 \,{\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )}}{b^{8} c d^{6} - a b^{7} d^{7}}\right )} \sqrt{b x + a}}{1536 \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}} - \frac{5 \,{\left (b c - a d\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{512 \, \sqrt{b d} b^{5} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)/(d*x + c)^(3/2),x, algorithm="giac")

[Out]

1/1536*((2*(b*x + a)*b^2*d^4/(b^8*c*d^6 - a*b^7*d^7) - 5*(b^3*c*d^3 - a*b^2*d^4)
/(b^8*c*d^6 - a*b^7*d^7))*(b*x + a) - 15*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*
d^4)/(b^8*c*d^6 - a*b^7*d^7))*sqrt(b*x + a)/sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
- 5/512*(b*c - a*d)*ln(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d
- a*b*d)))/(sqrt(b*d)*b^5*d^4)