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

Optimal. Leaf size=102 \[ \frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{5/2}}-\frac{2 \sqrt{a+b x}}{d^2 \sqrt{c+d x}}-\frac{2 c (a+b x)^{3/2}}{3 d (c+d x)^{3/2} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.145598, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{5/2}}-\frac{2 \sqrt{a+b x}}{d^2 \sqrt{c+d x}}-\frac{2 c (a+b x)^{3/2}}{3 d (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.8274, size = 92, normalized size = 0.9 \[ \frac{2 \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{d^{\frac{5}{2}}} + \frac{2 c \left (a + b x\right )^{\frac{3}{2}}}{3 d \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{2 \sqrt{a + b x}}{d^{2} \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.158675, size = 114, normalized size = 1.12 \[ \frac{\sqrt{b} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{d^{5/2}}+\frac{2 \sqrt{a+b x} (b c (3 c+4 d x)-a d (2 c+3 d x))}{3 d^2 (c+d x)^{3/2} (a d-b c)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.028, size = 442, normalized size = 4.3 \[{\frac{1}{ \left ( 3\,ad-3\,bc \right ){d}^{2}} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}ab{d}^{3}-3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{b}^{2}c{d}^{2}+6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xabc{d}^{2}-6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{b}^{2}{c}^{2}d+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) ab{c}^{2}d-3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{2}{c}^{3}-6\,xa{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+8\,xbcd\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-4\,acd\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+6\,b{c}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ) \sqrt{bx+a}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(3*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^3-3*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+6*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^2-6*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+3*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-3*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^3-6*x*a*d^2*((b*x+a)*
(d*x+c))^(1/2)*(b*d)^(1/2)+8*x*b*c*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-4*a*c*d
*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+6*b*c^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)
)*(b*x+a)^(1/2)/(b*d)^(1/2)/(a*d-b*c)/((b*x+a)*(d*x+c))^(1/2)/d^2/(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(sqrt(b*x + a)*x/(d*x + c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.364516, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b c^{3} - a c^{2} d +{\left (b c d^{2} - a d^{3}\right )} x^{2} + 2 \,{\left (b c^{2} d - a c d^{2}\right )} x\right )} \sqrt{\frac{b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{b}{d}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \,{\left (3 \, b c^{2} - 2 \, a c d +{\left (4 \, b c d - 3 \, a d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{6 \,{\left (b c^{3} d^{2} - a c^{2} d^{3} +{\left (b c d^{4} - a d^{5}\right )} x^{2} + 2 \,{\left (b c^{2} d^{3} - a c d^{4}\right )} x\right )}}, \frac{3 \,{\left (b c^{3} - a c^{2} d +{\left (b c d^{2} - a d^{3}\right )} x^{2} + 2 \,{\left (b c^{2} d - a c d^{2}\right )} x\right )} \sqrt{-\frac{b}{d}} \arctan \left (\frac{2 \, b d x + b c + a d}{2 \, \sqrt{b x + a} \sqrt{d x + c} d \sqrt{-\frac{b}{d}}}\right ) - 2 \,{\left (3 \, b c^{2} - 2 \, a c d +{\left (4 \, b c d - 3 \, a d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (b c^{3} d^{2} - a c^{2} d^{3} +{\left (b c d^{4} - a d^{5}\right )} x^{2} + 2 \,{\left (b c^{2} d^{3} - a c d^{4}\right )} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.256327, size = 317, normalized size = 3.11 \[ \frac{\frac{3 \, \sqrt{b d}{\left | b \right |}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{b^{5} c d^{4} - a b^{4} d^{5}} + \frac{\sqrt{b x + a}{\left (\frac{{\left (4 \, b^{5} c d^{2}{\left | b \right |} - 3 \, a b^{4} d^{3}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}} + \frac{3 \,{\left (b^{6} c^{2} d{\left | b \right |} - 2 \, a b^{5} c d^{2}{\left | b \right |} + a^{2} b^{4} d^{3}{\left | b \right |}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )}}{{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}}}{12 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(3*sqrt(b*d)*abs(b)*ln(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a
)*b*d - a*b*d)))/(b^5*c*d^4 - a*b^4*d^5) + sqrt(b*x + a)*((4*b^5*c*d^2*abs(b) -
3*a*b^4*d^3*abs(b))*(b*x + a)/(b^8*c^2*d^4 - 2*a*b^7*c*d^5 + a^2*b^6*d^6) + 3*(b
^6*c^2*d*abs(b) - 2*a*b^5*c*d^2*abs(b) + a^2*b^4*d^3*abs(b))/(b^8*c^2*d^4 - 2*a*
b^7*c*d^5 + a^2*b^6*d^6))/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2))/b