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

Optimal. Leaf size=162 \[ -\frac{a^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{3/2}}-\frac{b^{3/2} (b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{3/2}}-\frac{a (a+b x)^{3/2} \sqrt{c+d x}}{c x}+\frac{b \sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{c d} \]

[Out]

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

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Rubi [A]  time = 0.504948, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{a^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{3/2}}-\frac{b^{3/2} (b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{3/2}}-\frac{a (a+b x)^{3/2} \sqrt{c+d x}}{c x}+\frac{b \sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{c d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 50.0854, size = 144, normalized size = 0.89 \[ \frac{a^{\frac{3}{2}} \left (a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{c^{\frac{3}{2}}} - \frac{a \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x}}{c x} + \frac{b^{\frac{3}{2}} \left (5 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{d^{\frac{3}{2}}} + \frac{b \sqrt{a + b x} \sqrt{c + d x} \left (a d + b c\right )}{c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.460468, size = 192, normalized size = 1.19 \[ \frac{1}{2} \left (-\frac{a^{3/2} \log (x) (a d-5 b c)}{c^{3/2}}+\frac{a^{3/2} (a d-5 b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{c^{3/2}}+2 \sqrt{a+b x} \sqrt{c+d x} \left (\frac{b^2}{d}-\frac{a^2}{c x}\right )+\frac{b^{3/2} (5 a d-b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{d^{3/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.029, size = 320, normalized size = 2. \[{\frac{1}{2\,cxd}\sqrt{bx+a}\sqrt{dx+c} \left ( 5\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xa{b}^{2}cd\sqrt{ac}-\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) x{b}^{3}{c}^{2}\sqrt{ac}+\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) x{a}^{3}{d}^{2}\sqrt{bd}-5\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{a}^{2}bcd\sqrt{bd}+2\,x{b}^{2}c\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-2\,{a}^{2}d\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x+a)^(1/2)*(d*x+c)^(1/2)/c*(5*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^2*c*d*(a*c)^(1/2)-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^3*c^2*(a*c)^(1/2)+ln((a
*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x*a^3*d^2*(b*d)^(1/2)
-5*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x*a^2*b*c*d*(
b*d)^(1/2)+2*x*b^2*c*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-2*a^2*d*(a*
c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/x/(b*d)^(1
/2)/(a*c)^(1/2)/d

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.01438, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*((b^2*c^2 - 5*a*b*c*d)*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) + (5*a*b*c*d - a^2*d^2)*x*sqrt(a/c)*log((8*a^2*c^2 +
(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 + 4*(2*a*c^2 + (b*c^2 + a*c*d)*x)*sqrt(b*x
+ a)*sqrt(d*x + c)*sqrt(a/c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 4*(b^2*c*x - a^2*
d)*sqrt(b*x + a)*sqrt(d*x + c))/(c*d*x), -1/4*(2*(b^2*c^2 - 5*a*b*c*d)*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)))
+ (5*a*b*c*d - a^2*d^2)*x*sqrt(a/c)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2
*d^2)*x^2 + 4*(2*a*c^2 + (b*c^2 + a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(a/c
) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 4*(b^2*c*x - a^2*d)*sqrt(b*x + a)*sqrt(d*x +
c))/(c*d*x), -1/4*(2*(5*a*b*c*d - a^2*d^2)*x*sqrt(-a/c)*arctan(1/2*(2*a*c + (b*
c + a*d)*x)/(sqrt(b*x + a)*sqrt(d*x + c)*c*sqrt(-a/c))) + (b^2*c^2 - 5*a*b*c*d)*
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*(b^2*c*x - a^2*d)*sqrt(b*x + a)*sqrt(d*x + c))/(c*d*x), -1/2*((5*a*b*c*d - a^
2*d^2)*x*sqrt(-a/c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(b*x + a)*sqrt(d*x +
c)*c*sqrt(-a/c))) + (b^2*c^2 - 5*a*b*c*d)*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*(b^2*c*x - a^2*d)*sqrt(
b*x + a)*sqrt(d*x + c))/(c*d*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((b*x+a)**(5/2)/x**2/(d*x+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.602867, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x