Optimal. Leaf size=94 \[ -\frac{d \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{b^{5/2}}-\frac{\left (c+d x^3\right )^{3/2}}{3 b \left (a+b x^3\right )}+\frac{d \sqrt{c+d x^3}}{b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0812484, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {444, 47, 50, 63, 208} \[ -\frac{d \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{b^{5/2}}-\frac{\left (c+d x^3\right )^{3/2}}{3 b \left (a+b x^3\right )}+\frac{d \sqrt{c+d x^3}}{b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 444
Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^2 \left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^2} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(c+d x)^{3/2}}{(a+b x)^2} \, dx,x,x^3\right )\\ &=-\frac{\left (c+d x^3\right )^{3/2}}{3 b \left (a+b x^3\right )}+\frac{d \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{a+b x} \, dx,x,x^3\right )}{2 b}\\ &=\frac{d \sqrt{c+d x^3}}{b^2}-\frac{\left (c+d x^3\right )^{3/2}}{3 b \left (a+b x^3\right )}+\frac{(d (b c-a d)) \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^3\right )}{2 b^2}\\ &=\frac{d \sqrt{c+d x^3}}{b^2}-\frac{\left (c+d x^3\right )^{3/2}}{3 b \left (a+b x^3\right )}+\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x^3}\right )}{b^2}\\ &=\frac{d \sqrt{c+d x^3}}{b^2}-\frac{\left (c+d x^3\right )^{3/2}}{3 b \left (a+b x^3\right )}-\frac{d \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{b^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0226425, size = 54, normalized size = 0.57 \[ \frac{2 d \left (c+d x^3\right )^{5/2} \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};-\frac{b \left (d x^3+c\right )}{a d-b c}\right )}{15 (a d-b c)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.005, size = 466, normalized size = 5. \begin{align*}{\frac{ad-bc}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }\sqrt{d{x}^{3}+c}}+{\frac{2\,d}{3\,{b}^{2}}\sqrt{d{x}^{3}+c}}+{\frac{{\frac{i}{2}}\sqrt{2}}{d{b}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ( b{{\it \_Z}}^{3}+a \right ) }{\sqrt [3]{-{d}^{2}c}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-{d}^{2}c}+\sqrt [3]{-{d}^{2}c} \right ) } \right ){\frac{1}{\sqrt [3]{-{d}^{2}c}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-{d}^{2}c}} \right ) \left ( -3\,\sqrt [3]{-{d}^{2}c}+i\sqrt{3}\sqrt [3]{-{d}^{2}c} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-{d}^{2}c}+\sqrt [3]{-{d}^{2}c} \right ) } \right ){\frac{1}{\sqrt [3]{-{d}^{2}c}}}}} \left ( i\sqrt [3]{-{d}^{2}c}{\it \_alpha}\,\sqrt{3}d-i\sqrt{3} \left ( -{d}^{2}c \right ) ^{{\frac{2}{3}}}+2\,{{\it \_alpha}}^{2}{d}^{2}-\sqrt [3]{-{d}^{2}c}{\it \_alpha}\,d- \left ( -{d}^{2}c \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-{d}^{2}c}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-{d}^{2}c}} \right ){\frac{1}{\sqrt [3]{-{d}^{2}c}}}}}},{\frac{b}{2\,d \left ( ad-bc \right ) } \left ( 2\,i\sqrt [3]{-{d}^{2}c}\sqrt{3}{{\it \_alpha}}^{2}d-i \left ( -{d}^{2}c \right ) ^{{\frac{2}{3}}}\sqrt{3}{\it \_alpha}+i\sqrt{3}cd-3\, \left ( -{d}^{2}c \right ) ^{2/3}{\it \_alpha}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-{d}^{2}c} \left ( -{\frac{3}{2\,d}\sqrt [3]{-{d}^{2}c}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-{d}^{2}c}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \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 [A] time = 1.57281, size = 493, normalized size = 5.24 \begin{align*} \left [\frac{3 \,{\left (b d x^{3} + a d\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{3} + 2 \, b c - a d - 2 \, \sqrt{d x^{3} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{3} + a}\right ) + 2 \,{\left (2 \, b d x^{3} - b c + 3 \, a d\right )} \sqrt{d x^{3} + c}}{6 \,{\left (b^{3} x^{3} + a b^{2}\right )}}, -\frac{3 \,{\left (b d x^{3} + a d\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (-\frac{\sqrt{d x^{3} + c} b \sqrt{-\frac{b c - a d}{b}}}{b c - a d}\right ) -{\left (2 \, b d x^{3} - b c + 3 \, a d\right )} \sqrt{d x^{3} + c}}{3 \,{\left (b^{3} x^{3} + a b^{2}\right )}}\right ] \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 = 1.14556, size = 161, normalized size = 1.71 \begin{align*} \frac{1}{3} \, d{\left (\frac{3 \,{\left (b c - a d\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{2}} + \frac{2 \, \sqrt{d x^{3} + c}}{b^{2}} - \frac{\sqrt{d x^{3} + c} b c - \sqrt{d x^{3} + c} a d}{{\left ({\left (d x^{3} + c\right )} b - b c + a d\right )} b^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]