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

Optimal. Leaf size=163 \[ \frac{c^{3/2} (3 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2}}-\frac{\sqrt{c+d x} (3 b c-2 a d) (b c-a d)}{a^2 b \sqrt{a+b x}}+\frac{2 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2}}-\frac{c (c+d x)^{3/2}}{a x \sqrt{a+b x}} \]

[Out]

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

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Rubi [A]  time = 0.522031, antiderivative size = 163, 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 \[ \frac{c^{3/2} (3 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2}}-\frac{\sqrt{c+d x} (3 b c-2 a d) (b c-a d)}{a^2 b \sqrt{a+b x}}+\frac{2 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2}}-\frac{c (c+d x)^{3/2}}{a x \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 32.5743, size = 150, normalized size = 0.92 \[ \frac{2 d^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{b^{\frac{3}{2}}} - \frac{c \left (c + d x\right )^{\frac{3}{2}}}{a x \sqrt{a + b x}} - \frac{\sqrt{c + d x} \left (a d - b c\right ) \left (2 a d - 3 b c\right )}{a^{2} b \sqrt{a + b x}} - \frac{c^{\frac{3}{2}} \left (5 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.543031, size = 199, normalized size = 1.22 \[ \frac{1}{2} \left (\frac{c^{3/2} \log (x) (5 a d-3 b c)}{a^{5/2}}+\frac{c^{3/2} (3 b c-5 a d) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{a^{5/2}}-\frac{2 \sqrt{a+b x} \sqrt{c+d x} \left (\frac{2 (b c-a d)^2}{b (a+b x)}+\frac{c^2}{x}\right )}{a^2}+\frac{2 d^{5/2} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{b^{3/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.037, size = 502, normalized size = 3.1 \[ -{\frac{1}{2\,{a}^{2}xb}\sqrt{dx+c} \left ( 5\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}a{b}^{2}{c}^{2}d\sqrt{bd}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{b}^{3}{c}^{3}\sqrt{bd}-2\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{a}^{2}b{d}^{3}\sqrt{ac}+5\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{a}^{2}b{c}^{2}d\sqrt{bd}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) xa{b}^{2}{c}^{3}\sqrt{bd}-2\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{a}^{3}{d}^{3}\sqrt{ac}+4\,x{a}^{2}{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}-8\,xabcd\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}+6\,x{b}^{2}{c}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}+2\,ab{c}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(d*x+c)^(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^2*a*b^2*c^2*d*(b*d)^(1/2)-3*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^2*b^3*c^3*(b*d)^(1/2)-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^2*a^2*b*d^3*(a*c)^(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^2*d*(b*d)^(1/2)-
3*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*b^2*c^3*(b
*d)^(1/2)-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*a^3*d^3*(a*c)^(1/2)+4*x*a^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a
*c)^(1/2)-8*x*a*b*c*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)+6*x*b^2*c^
2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)+2*a*b*c^2*(a*c)^(1/2)*((b*x+a)
*(d*x+c))^(1/2)*(b*d)^(1/2))/a^2/((b*x+a)*(d*x+c))^(1/2)/x/(b*d)^(1/2)/(a*c)^(1/
2)/(b*x+a)^(1/2)/b

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.71938, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(2*(a^2*b*d^2*x^2 + a^3*d^2*x)*sqrt(d/b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*
b*c*d + a^2*d^2 + 4*(2*b^2*d*x + b^2*c + a*b*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt
(d/b) + 8*(b^2*c*d + a*b*d^2)*x) - ((3*b^3*c^2 - 5*a*b^2*c*d)*x^2 + (3*a*b^2*c^2
- 5*a^2*b*c*d)*x)*sqrt(c/a)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^
2 - 4*(2*a^2*c + (a*b*c + a^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(c/a) + 8*(a
*b*c^2 + a^2*c*d)*x)/x^2) - 4*(a*b*c^2 + (3*b^2*c^2 - 4*a*b*c*d + 2*a^2*d^2)*x)*
sqrt(b*x + a)*sqrt(d*x + c))/(a^2*b^2*x^2 + a^3*b*x), 1/4*(4*(a^2*b*d^2*x^2 + a^
3*d^2*x)*sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)/(sqrt(b*x + a)*sqrt(d*x + c
)*b*sqrt(-d/b))) - ((3*b^3*c^2 - 5*a*b^2*c*d)*x^2 + (3*a*b^2*c^2 - 5*a^2*b*c*d)*
x)*sqrt(c/a)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a^2*c +
(a*b*c + a^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(c/a) + 8*(a*b*c^2 + a^2*c*d
)*x)/x^2) - 4*(a*b*c^2 + (3*b^2*c^2 - 4*a*b*c*d + 2*a^2*d^2)*x)*sqrt(b*x + a)*sq
rt(d*x + c))/(a^2*b^2*x^2 + a^3*b*x), 1/2*(((3*b^3*c^2 - 5*a*b^2*c*d)*x^2 + (3*a
*b^2*c^2 - 5*a^2*b*c*d)*x)*sqrt(-c/a)*arctan(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(b
*x + a)*sqrt(d*x + c)*a*sqrt(-c/a))) + (a^2*b*d^2*x^2 + a^3*d^2*x)*sqrt(d/b)*log
(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b^2*d*x + b^2*c + a*b*d)*s
qrt(b*x + a)*sqrt(d*x + c)*sqrt(d/b) + 8*(b^2*c*d + a*b*d^2)*x) - 2*(a*b*c^2 + (
3*b^2*c^2 - 4*a*b*c*d + 2*a^2*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*b^2*x^2
+ a^3*b*x), 1/2*(((3*b^3*c^2 - 5*a*b^2*c*d)*x^2 + (3*a*b^2*c^2 - 5*a^2*b*c*d)*x)
*sqrt(-c/a)*arctan(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(b*x + a)*sqrt(d*x + c)*a*sq
rt(-c/a))) + 2*(a^2*b*d^2*x^2 + a^3*d^2*x)*sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c
+ a*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*sqrt(-d/b))) - 2*(a*b*c^2 + (3*b^2*c^2 - 4
*a*b*c*d + 2*a^2*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*b^2*x^2 + a^3*b*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((d*x+c)**(5/2)/x**2/(b*x+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.630736, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x