Optimal. Leaf size=160 \[ \frac{(b c-a d)^{3/2} (3 a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^2 b^{5/2}}-\frac{c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a^2}-\frac{d \sqrt{c+d x^2} (b c-3 a d)}{2 a b^2}+\frac{\left (c+d x^2\right )^{3/2} (b c-a d)}{2 a b \left (a+b x^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.218048, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {446, 98, 154, 156, 63, 208} \[ \frac{(b c-a d)^{3/2} (3 a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^2 b^{5/2}}-\frac{c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a^2}-\frac{d \sqrt{c+d x^2} (b c-3 a d)}{2 a b^2}+\frac{\left (c+d x^2\right )^{3/2} (b c-a d)}{2 a b \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 446
Rule 98
Rule 154
Rule 156
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (c+d x^2\right )^{5/2}}{x \left (a+b x^2\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(c+d x)^{5/2}}{x (a+b x)^2} \, dx,x,x^2\right )\\ &=\frac{(b c-a d) \left (c+d x^2\right )^{3/2}}{2 a b \left (a+b x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{c+d x} \left (b c^2-\frac{1}{2} d (b c-3 a d) x\right )}{x (a+b x)} \, dx,x,x^2\right )}{2 a b}\\ &=-\frac{d (b c-3 a d) \sqrt{c+d x^2}}{2 a b^2}+\frac{(b c-a d) \left (c+d x^2\right )^{3/2}}{2 a b \left (a+b x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{\frac{b^2 c^3}{2}+\frac{1}{4} d \left (b^2 c^2+4 a b c d-3 a^2 d^2\right ) x}{x (a+b x) \sqrt{c+d x}} \, dx,x,x^2\right )}{a b^2}\\ &=-\frac{d (b c-3 a d) \sqrt{c+d x^2}}{2 a b^2}+\frac{(b c-a d) \left (c+d x^2\right )^{3/2}}{2 a b \left (a+b x^2\right )}+\frac{c^3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,x^2\right )}{2 a^2}-\frac{\left ((b c-a d)^2 (2 b c+3 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^2\right )}{4 a^2 b^2}\\ &=-\frac{d (b c-3 a d) \sqrt{c+d x^2}}{2 a b^2}+\frac{(b c-a d) \left (c+d x^2\right )^{3/2}}{2 a b \left (a+b x^2\right )}+\frac{c^3 \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x^2}\right )}{a^2 d}-\frac{\left ((b c-a d)^2 (2 b c+3 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x^2}\right )}{2 a^2 b^2 d}\\ &=-\frac{d (b c-3 a d) \sqrt{c+d x^2}}{2 a b^2}+\frac{(b c-a d) \left (c+d x^2\right )^{3/2}}{2 a b \left (a+b x^2\right )}-\frac{c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a^2}+\frac{(b c-a d)^{3/2} (2 b c+3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^2 b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.196237, size = 158, normalized size = 0.99 \[ \frac{\frac{a \sqrt{c+d x^2} \left (3 a^2 d^2+2 a b d \left (d x^2-c\right )+b^2 c^2\right )}{b^2 \left (a+b x^2\right )}+\frac{\sqrt{b c-a d} \left (-3 a^2 d^2+a b c d+2 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{5/2}}-2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.016, size = 7477, normalized size = 46.7 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{2} + c\right )}^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{2} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 10.9943, size = 2377, normalized size = 14.86 \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 [A] time = 1.15474, size = 290, normalized size = 1.81 \begin{align*} \frac{1}{2} \, d^{2}{\left (\frac{2 \, c^{3} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} d^{2}} + \frac{2 \, \sqrt{d x^{2} + c}}{b^{2}} + \frac{\sqrt{d x^{2} + c} b^{2} c^{2} - 2 \, \sqrt{d x^{2} + c} a b c d + \sqrt{d x^{2} + c} a^{2} d^{2}}{{\left ({\left (d x^{2} + c\right )} b - b c + a d\right )} a b^{2} d} - \frac{{\left (2 \, b^{3} c^{3} - a b^{2} c^{2} d - 4 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} b^{2} d^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]