Optimal. Leaf size=442 \[ -\frac {2 a^2}{c e \sqrt {e x} \left (c+d x^2\right )^{3/2}}-\frac {\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) (e x)^{3/2}}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}+\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) (e x)^{3/2}}{2 c^3 d e^3 \sqrt {c+d x^2}}-\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{2 c^3 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 c^{11/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{4 c^{11/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.30, antiderivative size = 442, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {473, 468, 296,
335, 311, 226, 1210} \begin {gather*} -\frac {(e x)^{3/2} \left (7 a^2 d^2-2 a b c d+b^2 c^2\right )}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}-\frac {2 a^2}{c e \sqrt {e x} \left (c+d x^2\right )^{3/2}}-\frac {\left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (a d (2 b c-7 a d)+b^2 c^2\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{4 c^{11/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}}+\frac {\left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (a d (2 b c-7 a d)+b^2 c^2\right ) E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 c^{11/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {\sqrt {e x} \sqrt {c+d x^2} \left (a d (2 b c-7 a d)+b^2 c^2\right )}{2 c^3 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {(e x)^{3/2} \left (a d (2 b c-7 a d)+b^2 c^2\right )}{2 c^3 d e^3 \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 226
Rule 296
Rule 311
Rule 335
Rule 468
Rule 473
Rule 1210
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} \left (c+d x^2\right )^{5/2}} \, dx &=-\frac {2 a^2}{c e \sqrt {e x} \left (c+d x^2\right )^{3/2}}+\frac {2 \int \frac {\sqrt {e x} \left (\frac {1}{2} a (2 b c-7 a d)+\frac {1}{2} b^2 c x^2\right )}{\left (c+d x^2\right )^{5/2}} \, dx}{c e^2}\\ &=-\frac {2 a^2}{c e \sqrt {e x} \left (c+d x^2\right )^{3/2}}-\frac {\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) (e x)^{3/2}}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}+\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) \int \frac {\sqrt {e x}}{\left (c+d x^2\right )^{3/2}} \, dx}{2 c^2 d e^2}\\ &=-\frac {2 a^2}{c e \sqrt {e x} \left (c+d x^2\right )^{3/2}}-\frac {\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) (e x)^{3/2}}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}+\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) (e x)^{3/2}}{2 c^3 d e^3 \sqrt {c+d x^2}}-\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx}{4 c^3 d e^2}\\ &=-\frac {2 a^2}{c e \sqrt {e x} \left (c+d x^2\right )^{3/2}}-\frac {\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) (e x)^{3/2}}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}+\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) (e x)^{3/2}}{2 c^3 d e^3 \sqrt {c+d x^2}}-\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 c^3 d e^3}\\ &=-\frac {2 a^2}{c e \sqrt {e x} \left (c+d x^2\right )^{3/2}}-\frac {\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) (e x)^{3/2}}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}+\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) (e x)^{3/2}}{2 c^3 d e^3 \sqrt {c+d x^2}}-\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 c^{5/2} d^{3/2} e^2}+\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 c^{5/2} d^{3/2} e^2}\\ &=-\frac {2 a^2}{c e \sqrt {e x} \left (c+d x^2\right )^{3/2}}-\frac {\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) (e x)^{3/2}}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}+\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) (e x)^{3/2}}{2 c^3 d e^3 \sqrt {c+d x^2}}-\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{2 c^3 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 c^{11/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{4 c^{11/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.13, size = 161, normalized size = 0.36 \begin {gather*} \frac {x \left (b^2 c^2 x^2 \left (c+3 d x^2\right )+2 a b c d x^2 \left (5 c+3 d x^2\right )-a^2 d \left (12 c^2+35 c d x^2+21 d^2 x^4\right )-\left (b^2 c^2+2 a b c d-7 a^2 d^2\right ) x^2 \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {d x^2}{c}\right )\right )}{6 c^3 d (e x)^{3/2} \left (c+d x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1186\) vs.
\(2(452)=904\).
time = 0.17, size = 1187, normalized size = 2.69 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.26, size = 244, normalized size = 0.55 \begin {gather*} \frac {{\left (3 \, {\left ({\left (b^{2} c^{2} d^{2} + 2 \, a b c d^{3} - 7 \, a^{2} d^{4}\right )} x^{5} + 2 \, {\left (b^{2} c^{3} d + 2 \, a b c^{2} d^{2} - 7 \, a^{2} c d^{3}\right )} x^{3} + {\left (b^{2} c^{4} + 2 \, a b c^{3} d - 7 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt {d} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) - {\left (12 \, a^{2} c^{2} d^{2} - 3 \, {\left (b^{2} c^{2} d^{2} + 2 \, a b c d^{3} - 7 \, a^{2} d^{4}\right )} x^{4} - {\left (b^{2} c^{3} d + 10 \, a b c^{2} d^{2} - 35 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {x}\right )} e^{\left (-\frac {3}{2}\right )}}{6 \, {\left (c^{3} d^{4} x^{5} + 2 \, c^{4} d^{3} x^{3} + c^{5} d^{2} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x^{2}\right )^{2}}{\left (e x\right )^{\frac {3}{2}} \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (b\,x^2+a\right )}^2}{{\left (e\,x\right )}^{3/2}\,{\left (d\,x^2+c\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________