Optimal. Leaf size=554 \[ -\frac{385 d^7 (d x)^{3/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{55 d^5 (d x)^{7/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 d^{17/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} \sqrt [4]{a} b^{19/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1155 d^{17/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} \sqrt [4]{a} b^{19/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1155 d^{17/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} \sqrt [4]{a} b^{19/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 d^{17/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} \sqrt [4]{a} b^{19/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.419476, antiderivative size = 554, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1112, 288, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{385 d^7 (d x)^{3/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{55 d^5 (d x)^{7/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 d^{17/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} \sqrt [4]{a} b^{19/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1155 d^{17/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} \sqrt [4]{a} b^{19/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1155 d^{17/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} \sqrt [4]{a} b^{19/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 d^{17/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} \sqrt [4]{a} b^{19/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1112
Rule 288
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x^2\right )\right ) \int \frac{(d x)^{17/2}}{\left (a b+b^2 x^2\right )^5} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (15 b^2 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac{(d x)^{13/2}}{\left (a b+b^2 x^2\right )^4} \, dx}{16 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (55 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac{(d x)^{9/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{64 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{55 d^5 (d x)^{7/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (385 d^6 \left (a b+b^2 x^2\right )\right ) \int \frac{(d x)^{5/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{512 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{385 d^7 (d x)^{3/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{55 d^5 (d x)^{7/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (1155 d^8 \left (a b+b^2 x^2\right )\right ) \int \frac{\sqrt{d x}}{a b+b^2 x^2} \, dx}{2048 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{385 d^7 (d x)^{3/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{55 d^5 (d x)^{7/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (1155 d^7 \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 )}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{385 d^7 (d x)^{3/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{55 d^5 (d x)^{7/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (1155 d^7 \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 )}{2048 b^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (1155 d^7 \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 )}{2048 b^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{385 d^7 (d x)^{3/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{55 d^5 (d x)^{7/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (1155 d^{17/2} \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 )}{4096 \sqrt{2} \sqrt [4]{a} b^{23/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (1155 d^{17/2} \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 )}{4096 \sqrt{2} \sqrt [4]{a} b^{23/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (1155 d^9 \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 )}{4096 b^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (1155 d^9 \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 )}{4096 b^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{385 d^7 (d x)^{3/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{55 d^5 (d x)^{7/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 d^{17/2} \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 )}{4096 \sqrt{2} \sqrt [4]{a} b^{19/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1155 d^{17/2} \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 )}{4096 \sqrt{2} \sqrt [4]{a} b^{19/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (1155 d^{17/2} \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 )}{2048 \sqrt{2} \sqrt [4]{a} b^{23/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (1155 d^{17/2} \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 )}{2048 \sqrt{2} \sqrt [4]{a} b^{23/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{385 d^7 (d x)^{3/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{15/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{55 d^5 (d x)^{7/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1155 d^{17/2} \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 )}{2048 \sqrt{2} \sqrt [4]{a} b^{19/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 d^{17/2} \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 )}{2048 \sqrt{2} \sqrt [4]{a} b^{19/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 d^{17/2} \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 )}{4096 \sqrt{2} \sqrt [4]{a} b^{19/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1155 d^{17/2} \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 )}{4096 \sqrt{2} \sqrt [4]{a} b^{19/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0463936, size = 106, normalized size = 0.19 \[ \frac{2 d^7 (d x)^{3/2} \left (77 \left (a+b x^2\right )^4 \, _2F_1\left (\frac{3}{4},5;\frac{7}{4};-\frac{b x^2}{a}\right )-a \left (143 a^2 b x^2+77 a^3+117 a b^2 x^4+39 b^3 x^6\right )\right )}{39 a b^4 \left (a+b x^2\right )^3 \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.234, size = 1046, normalized size = 1.9 \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 = 1.70759, size = 992, normalized size = 1.79 \begin{align*} -\frac{4620 \,{\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )} \left (-\frac{d^{34}}{a b^{19}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\left (-\frac{d^{34}}{a b^{19}}\right )^{\frac{1}{4}} \sqrt{d x} b^{5} d^{25} - \sqrt{d^{51} x - \sqrt{-\frac{d^{34}}{a b^{19}}} a b^{9} d^{34}} \left (-\frac{d^{34}}{a b^{19}}\right )^{\frac{1}{4}} b^{5}}{d^{34}}\right ) - 1155 \,{\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )} \left (-\frac{d^{34}}{a b^{19}}\right )^{\frac{1}{4}} \log \left (1540798875 \, \sqrt{d x} d^{25} + 1540798875 \, \left (-\frac{d^{34}}{a b^{19}}\right )^{\frac{3}{4}} a b^{14}\right ) + 1155 \,{\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )} \left (-\frac{d^{34}}{a b^{19}}\right )^{\frac{1}{4}} \log \left (1540798875 \, \sqrt{d x} d^{25} - 1540798875 \, \left (-\frac{d^{34}}{a b^{19}}\right )^{\frac{3}{4}} a b^{14}\right ) + 4 \,{\left (893 \, b^{3} d^{8} x^{7} + 1755 \, a b^{2} d^{8} x^{5} + 1375 \, a^{2} b d^{8} x^{3} + 385 \, a^{3} d^{8} x\right )} \sqrt{d x}}{4096 \,{\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\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 = 1.43693, size = 548, normalized size = 0.99 \begin{align*} \frac{1}{8192} \, d^{7}{\left (\frac{2310 \, \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 )}{a b^{7} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{2310 \, \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 )}{a b^{7} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{1155 \, \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 )}{a b^{7} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{1155 \, \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 )}{a b^{7} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{8 \,{\left (893 \, \sqrt{d x} b^{3} d^{9} x^{7} + 1755 \, \sqrt{d x} a b^{2} d^{9} x^{5} + 1375 \, \sqrt{d x} a^{2} b d^{9} x^{3} + 385 \, \sqrt{d x} a^{3} d^{9} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} b^{4} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]