Optimal. Leaf size=553 \[ \frac{117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{117 b^{5/4} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{17/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{117 b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{17/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{117 b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{17/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{117 b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} a^{17/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{13}{16 a^2 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.432307, antiderivative size = 553, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1112, 290, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{117 b^{5/4} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{17/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{117 b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{17/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{117 b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{17/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{117 b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} a^{17/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{13}{16 a^2 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1112
Rule 290
Rule 325
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^3} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (13 b \left (a b+b^2 x^2\right )\right ) \int \frac{1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^2} \, dx}{8 a \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{13}{16 a^2 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (117 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{(d x)^{7/2} \left (a b+b^2 x^2\right )} \, dx}{32 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{13}{16 a^2 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (117 b \left (a b+b^2 x^2\right )\right ) \int \frac{1}{(d x)^{3/2} \left (a b+b^2 x^2\right )} \, dx}{32 a^3 d^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{13}{16 a^2 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (117 b^2 \left (a b+b^2 x^2\right )\right ) \int \frac{\sqrt{d x}}{a b+b^2 x^2} \, dx}{32 a^4 d^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{13}{16 a^2 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (117 b^2 \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{16 a^4 d^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{13}{16 a^2 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (117 b^{3/2} \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} d-\sqrt{b} x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{32 a^4 d^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (117 b^{3/2} \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} d+\sqrt{b} x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{32 a^4 d^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{13}{16 a^2 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (117 \sqrt [4]{b} \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a} d}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{d x}\right )}{64 \sqrt{2} a^{17/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (117 \sqrt [4]{b} \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a} d}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{d x}\right )}{64 \sqrt{2} a^{17/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (117 \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a} d}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{d x}\right )}{64 a^4 d^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (117 \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a} d}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{d x}\right )}{64 a^4 d^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{13}{16 a^2 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{117 b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{64 \sqrt{2} a^{17/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{117 b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{64 \sqrt{2} a^{17/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (117 \sqrt [4]{b} \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{17/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (117 \sqrt [4]{b} \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{17/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{13}{16 a^2 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d (d x)^{5/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{117 \left (a+b x^2\right )}{80 a^3 d (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{117 b \left (a+b x^2\right )}{16 a^4 d^3 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{117 b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{17/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{117 b^{5/4} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{17/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{117 b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{64 \sqrt{2} a^{17/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{117 b^{5/4} \left (a+b x^2\right ) \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{64 \sqrt{2} a^{17/4} d^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0160301, size = 54, normalized size = 0.1 \[ -\frac{2 x \left (a+b x^2\right )^3 \, _2F_1\left (-\frac{5}{4},3;-\frac{1}{4};-\frac{b x^2}{a}\right )}{5 a^3 (d x)^{7/2} \left (\left (a+b x^2\right )^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.247, size = 687, normalized size = 1.2 \begin{align*} \text{result too large to display} \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 = 1.69873, size = 961, normalized size = 1.74 \begin{align*} -\frac{2340 \,{\left (a^{4} b^{2} d^{4} x^{7} + 2 \, a^{5} b d^{4} x^{5} + a^{6} d^{4} x^{3}\right )} \left (-\frac{b^{5}}{a^{17} d^{14}}\right )^{\frac{1}{4}} \arctan \left (-\frac{1601613 \, \sqrt{d x} a^{4} b^{4} d^{3} \left (-\frac{b^{5}}{a^{17} d^{14}}\right )^{\frac{1}{4}} - \sqrt{-2565164201769 \, a^{9} b^{5} d^{8} \sqrt{-\frac{b^{5}}{a^{17} d^{14}}} + 2565164201769 \, b^{8} d x} a^{4} d^{3} \left (-\frac{b^{5}}{a^{17} d^{14}}\right )^{\frac{1}{4}}}{1601613 \, b^{5}}\right ) - 585 \,{\left (a^{4} b^{2} d^{4} x^{7} + 2 \, a^{5} b d^{4} x^{5} + a^{6} d^{4} x^{3}\right )} \left (-\frac{b^{5}}{a^{17} d^{14}}\right )^{\frac{1}{4}} \log \left (1601613 \, a^{13} d^{11} \left (-\frac{b^{5}}{a^{17} d^{14}}\right )^{\frac{3}{4}} + 1601613 \, \sqrt{d x} b^{4}\right ) + 585 \,{\left (a^{4} b^{2} d^{4} x^{7} + 2 \, a^{5} b d^{4} x^{5} + a^{6} d^{4} x^{3}\right )} \left (-\frac{b^{5}}{a^{17} d^{14}}\right )^{\frac{1}{4}} \log \left (-1601613 \, a^{13} d^{11} \left (-\frac{b^{5}}{a^{17} d^{14}}\right )^{\frac{3}{4}} + 1601613 \, \sqrt{d x} b^{4}\right ) - 4 \,{\left (585 \, b^{3} x^{6} + 1053 \, a b^{2} x^{4} + 416 \, a^{2} b x^{2} - 32 \, a^{3}\right )} \sqrt{d x}}{320 \,{\left (a^{4} b^{2} d^{4} x^{7} + 2 \, a^{5} b d^{4} x^{5} + a^{6} d^{4} x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d x\right )^{\frac{7}{2}} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.33656, size = 583, normalized size = 1.05 \begin{align*} \frac{21 \, \sqrt{d x} b^{3} d^{3} x^{3} + 25 \, \sqrt{d x} a b^{2} d^{3} x}{16 \,{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} a^{4} d^{3} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{117 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{5} b d^{5} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{117 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{5} b d^{5} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{117 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \log \left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{128 \, a^{5} b d^{5} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{117 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \log \left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{128 \, a^{5} b d^{5} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{2 \,{\left (15 \, b d^{2} x^{2} - a d^{2}\right )}}{5 \, \sqrt{d x} a^{4} d^{5} x^{2} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]