Optimal. Leaf size=157 \[ \frac{2 \sqrt{c+d x} \left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right )}{\sqrt{a+b x}}-\frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2}}+\frac{2 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2}}+\frac{2 (c+d x)^{3/2} (b c-a d)}{3 a b (a+b x)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.111729, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {98, 150, 157, 63, 217, 206, 93, 208} \[ \frac{2 \sqrt{c+d x} \left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right )}{\sqrt{a+b x}}-\frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2}}+\frac{2 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2}}+\frac{2 (c+d x)^{3/2} (b c-a d)}{3 a b (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 98
Rule 150
Rule 157
Rule 63
Rule 217
Rule 206
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(c+d x)^{5/2}}{x (a+b x)^{5/2}} \, dx &=\frac{2 (b c-a d) (c+d x)^{3/2}}{3 a b (a+b x)^{3/2}}+\frac{2 \int \frac{\sqrt{c+d x} \left (\frac{3 b c^2}{2}+\frac{3}{2} a d^2 x\right )}{x (a+b x)^{3/2}} \, dx}{3 a b}\\ &=\frac{2 \left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right ) \sqrt{c+d x}}{\sqrt{a+b x}}+\frac{2 (b c-a d) (c+d x)^{3/2}}{3 a b (a+b x)^{3/2}}-\frac{4 \int \frac{-\frac{3}{4} b^2 c^3-\frac{3}{4} a^2 d^3 x}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{3 a^2 b^2}\\ &=\frac{2 \left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right ) \sqrt{c+d x}}{\sqrt{a+b x}}+\frac{2 (b c-a d) (c+d x)^{3/2}}{3 a b (a+b x)^{3/2}}+\frac{c^3 \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{a^2}+\frac{d^3 \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{b^2}\\ &=\frac{2 \left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right ) \sqrt{c+d x}}{\sqrt{a+b x}}+\frac{2 (b c-a d) (c+d x)^{3/2}}{3 a b (a+b x)^{3/2}}+\frac{\left (2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{a^2}+\frac{\left (2 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{b^3}\\ &=\frac{2 \left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right ) \sqrt{c+d x}}{\sqrt{a+b x}}+\frac{2 (b c-a d) (c+d x)^{3/2}}{3 a b (a+b x)^{3/2}}-\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{\left (2 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{b^3}\\ &=\frac{2 \left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right ) \sqrt{c+d x}}{\sqrt{a+b x}}+\frac{2 (b c-a d) (c+d x)^{3/2}}{3 a b (a+b x)^{3/2}}-\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 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.373722, size = 163, normalized size = 1.04 \[ \frac{2 \left (-\frac{3 c^{5/2} (a+b x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2}}+\frac{c \sqrt{c+d x} (4 a c+a d x+3 b c x)}{a^2}+\frac{d \sqrt{c+d x} (a d-b c) \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{d (a+b x)}{a d-b c}\right )}{b^2 \sqrt{\frac{b (c+d x)}{b c-a d}}}\right )}{3 (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.021, size = 566, normalized size = 3.6 \begin{align*}{\frac{1}{3\,{b}^{2}{a}^{2}}\sqrt{dx+c} \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}{a}^{2}{b}^{2}{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 ){x}^{2}{b}^{4}{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}-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}+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}-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}-8\,x{a}^{2}b{d}^{2}\sqrt{bd}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,xa{b}^{2}cd\sqrt{bd}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+6\,x{b}^{3}{c}^{2}\sqrt{bd}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-6\,{a}^{3}{d}^{2}\sqrt{bd}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-2\,{a}^{2}bcd\sqrt{bd}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+8\,a{b}^{2}{c}^{2}\sqrt{bd}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) } \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}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 14.5061, size = 2917, normalized size = 18.58 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 3.84131, size = 956, normalized size = 6.09 \begin{align*} -\frac{2 \, \sqrt{b d} c^{3}{\left | b \right |} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} a^{2} b} - \frac{\sqrt{b d} d^{2}{\left | b \right |} \log \left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{b^{4}} + \frac{4 \,{\left (3 \, \sqrt{b d} b^{7} c^{5}{\left | b \right |} - 8 \, \sqrt{b d} a b^{6} c^{4} d{\left | b \right |} + 2 \, \sqrt{b d} a^{2} b^{5} c^{3} d^{2}{\left | b \right |} + 12 \, \sqrt{b d} a^{3} b^{4} c^{2} d^{3}{\left | b \right |} - 13 \, \sqrt{b d} a^{4} b^{3} c d^{4}{\left | b \right |} + 4 \, \sqrt{b d} a^{5} b^{2} d^{5}{\left | b \right |} - 6 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} b^{5} c^{4}{\left | b \right |} + 12 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{4} c^{3} d{\left | b \right |} - 12 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{2} c d^{3}{\left | b \right |} + 6 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a^{4} b d^{4}{\left | b \right |} + 3 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} b^{3} c^{3}{\left | b \right |} - 9 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b c d^{2}{\left | b \right |} + 6 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} d^{3}{\left | b \right |}\right )}}{3 \,{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3} a^{2} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]