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

Optimal. Leaf size=174 \[ -\frac{\sqrt{b} (5 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{7/2}}+\frac{b \sqrt{a+b x} \sqrt{c+d x} (5 b c-3 a d)}{d^3 (b c-a d)}-\frac{2 (a+b x)^{3/2} (5 b c-3 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)}-\frac{2 c (a+b x)^{5/2}}{3 d (c+d x)^{3/2} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.234527, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sqrt{b} (5 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{7/2}}+\frac{b \sqrt{a+b x} \sqrt{c+d x} (5 b c-3 a d)}{d^3 (b c-a d)}-\frac{2 (a+b x)^{3/2} (5 b c-3 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)}-\frac{2 c (a+b x)^{5/2}}{3 d (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.685, size = 158, normalized size = 0.91 \[ \frac{\sqrt{b} \left (3 a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{d^{\frac{7}{2}}} + \frac{b \sqrt{a + b x} \sqrt{c + d x} \left (3 a d - 5 b c\right )}{d^{3} \left (a d - b c\right )} + \frac{2 c \left (a + b x\right )^{\frac{5}{2}}}{3 d \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (3 a d - 5 b c\right )}{3 d^{2} \sqrt{c + d x} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.291638, size = 126, normalized size = 0.72 \[ \frac{\sqrt{a+b x} \left (b \left (15 c^2+20 c d x+3 d^2 x^2\right )-2 a d (2 c+3 d x)\right )}{3 d^3 (c+d x)^{3/2}}+\frac{\sqrt{b} (3 a d-5 b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*(-2*a*d*(2*c + 3*d*x) + b*(15*c^2 + 20*c*d*x + 3*d^2*x^2)))/(3*d^
3*(c + d*x)^(3/2)) + (Sqrt[b]*(-5*b*c + 3*a*d)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[
b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(2*d^(7/2))

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Maple [B]  time = 0.03, size = 459, normalized size = 2.6 \[{\frac{1}{6\,{d}^{3}}\sqrt{bx+a} \left ( 9\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}ab{d}^{3}-15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{b}^{2}c{d}^{2}+18\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xabc{d}^{2}-30\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{b}^{2}{c}^{2}d+6\,{x}^{2}b{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+9\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) ab{c}^{2}d-15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{2}{c}^{3}-12\,xa{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+40\,xbcd\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-8\,acd\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+30\,b{c}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(b*x+a)^(1/2)*(9*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b
*c)/(b*d)^(1/2))*x^2*a*b*d^3-15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^
(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*b^2*c*d^2+18*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))
^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a*b*c*d^2-30*ln(1/2*(2*b*d*x+2*((b*x+
a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*b^2*c^2*d+6*x^2*b*d^2*((b*
x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+9*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d
)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*b*c^2*d-15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(
1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b^2*c^3-12*x*a*d^2*((b*x+a)*(d*x+c))^(1/2
)*(b*d)^(1/2)+40*x*b*c*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-8*a*c*d*((b*x+a)*(d
*x+c))^(1/2)*(b*d)^(1/2)+30*b*c^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/(b*d)^(1/
2)/((b*x+a)*(d*x+c))^(1/2)/d^3/(d*x+c)^(3/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.49561, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (5 \, b c^{3} - 3 \, a c^{2} d +{\left (5 \, b c d^{2} - 3 \, a d^{3}\right )} x^{2} + 2 \,{\left (5 \, b c^{2} d - 3 \, a c d^{2}\right )} x\right )} \sqrt{\frac{b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{b}{d}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \,{\left (3 \, b d^{2} x^{2} + 15 \, b c^{2} - 4 \, a c d + 2 \,{\left (10 \, b c d - 3 \, a d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{12 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}}, -\frac{3 \,{\left (5 \, b c^{3} - 3 \, a c^{2} d +{\left (5 \, b c d^{2} - 3 \, a d^{3}\right )} x^{2} + 2 \,{\left (5 \, b c^{2} d - 3 \, a c d^{2}\right )} x\right )} \sqrt{-\frac{b}{d}} \arctan \left (\frac{2 \, b d x + b c + a d}{2 \, \sqrt{b x + a} \sqrt{d x + c} d \sqrt{-\frac{b}{d}}}\right ) - 2 \,{\left (3 \, b d^{2} x^{2} + 15 \, b c^{2} - 4 \, a c d + 2 \,{\left (10 \, b c d - 3 \, a d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{6 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/12*(3*(5*b*c^3 - 3*a*c^2*d + (5*b*c*d^2 - 3*a*d^3)*x^2 + 2*(5*b*c^2*d - 3*a*
c*d^2)*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*(3*b*d^2*x^2 + 15*b*c^2 - 4*a*c*d + 2*(10*b*c*d - 3*a*d^2)*x)*sqrt(b*
x + a)*sqrt(d*x + c))/(d^5*x^2 + 2*c*d^4*x + c^2*d^3), -1/6*(3*(5*b*c^3 - 3*a*c^
2*d + (5*b*c*d^2 - 3*a*d^3)*x^2 + 2*(5*b*c^2*d - 3*a*c*d^2)*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*(3*b*
d^2*x^2 + 15*b*c^2 - 4*a*c*d + 2*(10*b*c*d - 3*a*d^2)*x)*sqrt(b*x + a)*sqrt(d*x
+ c))/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.252209, size = 386, normalized size = 2.22 \[ \frac{{\left ({\left (b x + a\right )}{\left (\frac{3 \,{\left (b^{5} c d^{4}{\left | b \right |} - a b^{4} d^{5}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{4} c d^{5} - a b^{3} d^{6}} + \frac{4 \,{\left (5 \, b^{6} c^{2} d^{3}{\left | b \right |} - 8 \, a b^{5} c d^{4}{\left | b \right |} + 3 \, a^{2} b^{4} d^{5}{\left | b \right |}\right )}}{b^{4} c d^{5} - a b^{3} d^{6}}\right )} + \frac{3 \,{\left (5 \, b^{7} c^{3} d^{2}{\left | b \right |} - 13 \, a b^{6} c^{2} d^{3}{\left | b \right |} + 11 \, a^{2} b^{5} c d^{4}{\left | b \right |} - 3 \, a^{3} b^{4} d^{5}{\left | b \right |}\right )}}{b^{4} c d^{5} - a b^{3} d^{6}}\right )} \sqrt{b x + a}}{3 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} + \frac{{\left (5 \, b c{\left | b \right |} - 3 \, a d{\left | b \right |}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*((b*x + a)*(3*(b^5*c*d^4*abs(b) - a*b^4*d^5*abs(b))*(b*x + a)/(b^4*c*d^5 - a
*b^3*d^6) + 4*(5*b^6*c^2*d^3*abs(b) - 8*a*b^5*c*d^4*abs(b) + 3*a^2*b^4*d^5*abs(b
))/(b^4*c*d^5 - a*b^3*d^6)) + 3*(5*b^7*c^3*d^2*abs(b) - 13*a*b^6*c^2*d^3*abs(b)
+ 11*a^2*b^5*c*d^4*abs(b) - 3*a^3*b^4*d^5*abs(b))/(b^4*c*d^5 - a*b^3*d^6))*sqrt(
b*x + a)/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) + (5*b*c*abs(b) - 3*a*d*abs(b))*l
n(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d
)*d^3)