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

Optimal. Leaf size=157 \[ -\frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2}}+\frac{2 \sqrt{c+d x} \left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right )}{\sqrt{a+b x}}+\frac{2 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2}}+\frac{2 (c+d x)^{3/2} (b c-a d)}{3 a b (a+b x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.419154, antiderivative size = 157, 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 \[ -\frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2}}+\frac{2 \sqrt{c+d x} \left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right )}{\sqrt{a+b x}}+\frac{2 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2}}+\frac{2 (c+d x)^{3/2} (b c-a d)}{3 a b (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 46.8722, size = 144, normalized size = 0.92 \[ \frac{\sqrt{c + d x} \left (- \frac{2 d^{2}}{b^{2}} + \frac{2 c^{2}}{a^{2}}\right )}{\sqrt{a + b x}} + \frac{2 d^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{b^{\frac{5}{2}}} - \frac{2 \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )}{3 a b \left (a + b x\right )^{\frac{3}{2}}} - \frac{2 c^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.6016, size = 183, normalized size = 1.17 \[ -\frac{c^{5/2} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{a^{5/2}}+\frac{c^{5/2} \log (x)}{a^{5/2}}+\frac{2 \sqrt{c+d x} (b c-a d) \left (3 a^2 d+4 a b (c+d x)+3 b^2 c x\right )}{3 a^2 b^2 (a+b x)^{3/2}}+\frac{d^{5/2} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.037, size = 566, normalized size = 3.6 \[ -{\frac{1}{3\,{a}^{2}{b}^{2}}\sqrt{dx+c} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{b}^{4}{c}^{3}\sqrt{bd}-3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{a}^{2}{b}^{2}{d}^{3}\sqrt{ac}+6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) xa{b}^{3}{c}^{3}\sqrt{bd}-6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{a}^{3}b{d}^{3}\sqrt{ac}+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){a}^{2}{b}^{2}{c}^{3}\sqrt{bd}-3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{4}{d}^{3}\sqrt{ac}+8\,x{a}^{2}b{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}-2\,xa{b}^{2}cd\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}-6\,x{b}^{3}{c}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}+6\,{a}^{3}{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}+2\,{a}^{2}bcd\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}-8\,a{b}^{2}{c}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.72789, 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)^(5/2)*x),x, algorithm="fricas")

[Out]

[1/6*(3*(a^2*b^2*d^2*x^2 + 2*a^3*b*d^2*x + a^4*d^2)*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)*sq
rt(d*x + c)*sqrt(d/b) + 8*(b^2*c*d + a*b*d^2)*x) + 3*(b^4*c^2*x^2 + 2*a*b^3*c^2*
x + a^2*b^2*c^2)*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*(4*a*b^2*c^2 - a^2*b*c*d - 3*a^3*d^2 + (3*b^3*c^2 +
a*b^2*c*d - 4*a^2*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*b^4*x^2 + 2*a^3*b^
3*x + a^4*b^2), 1/6*(6*(a^2*b^2*d^2*x^2 + 2*a^3*b*d^2*x + a^4*d^2)*sqrt(-d/b)*ar
ctan(1/2*(2*b*d*x + b*c + a*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*sqrt(-d/b))) + 3*(
b^4*c^2*x^2 + 2*a*b^3*c^2*x + a^2*b^2*c^2)*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*(4*a*b^2*c^2 - a^2*b*c*d -
3*a^3*d^2 + (3*b^3*c^2 + a*b^2*c*d - 4*a^2*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c
))/(a^2*b^4*x^2 + 2*a^3*b^3*x + a^4*b^2), -1/6*(6*(b^4*c^2*x^2 + 2*a*b^3*c^2*x +
a^2*b^2*c^2)*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))) - 3*(a^2*b^2*d^2*x^2 + 2*a^3*b*d^2*x + a^4*d^2)*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) - 4*(4*a*b^
2*c^2 - a^2*b*c*d - 3*a^3*d^2 + (3*b^3*c^2 + a*b^2*c*d - 4*a^2*b*d^2)*x)*sqrt(b*
x + a)*sqrt(d*x + c))/(a^2*b^4*x^2 + 2*a^3*b^3*x + a^4*b^2), -1/3*(3*(b^4*c^2*x^
2 + 2*a*b^3*c^2*x + a^2*b^2*c^2)*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))) - 3*(a^2*b^2*d^2*x^2 + 2*a^3*b*d^2*x
+ a^4*d^2)*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*(4*a*b^2*c^2 - a^2*b*c*d - 3*a^3*d^2 + (3*b^3*c^2 + a*b^2
*c*d - 4*a^2*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*b^4*x^2 + 2*a^3*b^3*x +
a^4*b^2)]

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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/(b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.649923, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x