Optimal. Leaf size=299 \[ \frac{2 a d (f x)^{5/2} \sqrt{a+b x^2+c x^4} F_1\left (\frac{5}{4};-\frac{3}{2},-\frac{3}{2};\frac{9}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{5 f \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}+\frac{2 a e (f x)^{9/2} \sqrt{a+b x^2+c x^4} F_1\left (\frac{9}{4};-\frac{3}{2},-\frac{3}{2};\frac{13}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{9 f^3 \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.347235, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {1335, 1141, 510} \[ \frac{2 a d (f x)^{5/2} \sqrt{a+b x^2+c x^4} F_1\left (\frac{5}{4};-\frac{3}{2},-\frac{3}{2};\frac{9}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{5 f \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}+\frac{2 a e (f x)^{9/2} \sqrt{a+b x^2+c x^4} F_1\left (\frac{9}{4};-\frac{3}{2},-\frac{3}{2};\frac{13}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{9 f^3 \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1335
Rule 1141
Rule 510
Rubi steps
\begin{align*} \int (f x)^{3/2} \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2} \, dx &=\int \left (d (f x)^{3/2} \left (a+b x^2+c x^4\right )^{3/2}+\frac{e (f x)^{7/2} \left (a+b x^2+c x^4\right )^{3/2}}{f^2}\right ) \, dx\\ &=d \int (f x)^{3/2} \left (a+b x^2+c x^4\right )^{3/2} \, dx+\frac{e \int (f x)^{7/2} \left (a+b x^2+c x^4\right )^{3/2} \, dx}{f^2}\\ &=\frac{\left (a d \sqrt{a+b x^2+c x^4}\right ) \int (f x)^{3/2} \left (1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )^{3/2} \left (1+\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )^{3/2} \, dx}{\sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}} \sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}}}+\frac{\left (a e \sqrt{a+b x^2+c x^4}\right ) \int (f x)^{7/2} \left (1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )^{3/2} \left (1+\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )^{3/2} \, dx}{f^2 \sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}} \sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}}}\\ &=\frac{2 a d (f x)^{5/2} \sqrt{a+b x^2+c x^4} F_1\left (\frac{5}{4};-\frac{3}{2},-\frac{3}{2};\frac{9}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{5 f \sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}} \sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}}}+\frac{2 a e (f x)^{9/2} \sqrt{a+b x^2+c x^4} F_1\left (\frac{9}{4};-\frac{3}{2},-\frac{3}{2};\frac{13}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{9 f^3 \sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}} \sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}}}\\ \end{align*}
Mathematica [F] time = 0, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{{\frac{3}{2}}} \left ( e{x}^{2}+d \right ) \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}\, dx \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{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}{\left (e x^{2} + d\right )} \left (f x\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c e f x^{7} +{\left (c d + b e\right )} f x^{5} +{\left (b d + a e\right )} f x^{3} + a d f x\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{f x}, x\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}{\left (e x^{2} + d\right )} \left (f x\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]