Optimal. Leaf size=375 \[ \frac {2 \sqrt {e x} \sqrt {c+d x^2} \left (15 a^2 d^2+b c (7 b c-18 a d)\right )}{15 d^{5/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (15 a^2 d^2+b c (7 b c-18 a d)\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{11/4} \sqrt {c+d x^2}}-\frac {2 \sqrt [4]{c} \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (15 a^2 d^2+b c (7 b c-18 a d)\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{11/4} \sqrt {c+d x^2}}-\frac {2 b (e x)^{3/2} \sqrt {c+d x^2} (7 b c-18 a d)}{45 d^2 e}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d e^3} \]
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Rubi [A] time = 0.36, antiderivative size = 375, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {464, 459, 329, 305, 220, 1196} \[ \frac {2 \sqrt {e x} \sqrt {c+d x^2} \left (15 a^2 d^2+b c (7 b c-18 a d)\right )}{15 d^{5/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (15 a^2 d^2+b c (7 b c-18 a d)\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{11/4} \sqrt {c+d x^2}}-\frac {2 \sqrt [4]{c} \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (15 a^2 d^2+b c (7 b c-18 a d)\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{11/4} \sqrt {c+d x^2}}-\frac {2 b (e x)^{3/2} \sqrt {c+d x^2} (7 b c-18 a d)}{45 d^2 e}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d e^3} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 329
Rule 459
Rule 464
Rule 1196
Rubi steps
\begin {align*} \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx &=\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d e^3}+\frac {2 \int \frac {\sqrt {e x} \left (\frac {9 a^2 d}{2}-\frac {1}{2} b (7 b c-18 a d) x^2\right )}{\sqrt {c+d x^2}} \, dx}{9 d}\\ &=-\frac {2 b (7 b c-18 a d) (e x)^{3/2} \sqrt {c+d x^2}}{45 d^2 e}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d e^3}+\frac {1}{15} \left (15 a^2+\frac {b c (7 b c-18 a d)}{d^2}\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx\\ &=-\frac {2 b (7 b c-18 a d) (e x)^{3/2} \sqrt {c+d x^2}}{45 d^2 e}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d e^3}+\frac {\left (2 \left (15 a^2+\frac {b c (7 b c-18 a d)}{d^2}\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 e}\\ &=-\frac {2 b (7 b c-18 a d) (e x)^{3/2} \sqrt {c+d x^2}}{45 d^2 e}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d e^3}+\frac {\left (2 \sqrt {c} \left (15 a^2+\frac {b c (7 b c-18 a d)}{d^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 \sqrt {d}}-\frac {\left (2 \sqrt {c} \left (15 a^2+\frac {b c (7 b c-18 a d)}{d^2}\right )\right ) \operatorname {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 )}{15 \sqrt {d}}\\ &=-\frac {2 b (7 b c-18 a d) (e x)^{3/2} \sqrt {c+d x^2}}{45 d^2 e}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d e^3}+\frac {2 \left (15 a^2+\frac {b c (7 b c-18 a d)}{d^2}\right ) \sqrt {e x} \sqrt {c+d x^2}}{15 \sqrt {d} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 \sqrt [4]{c} \left (15 a^2+\frac {b c (7 b c-18 a d)}{d^2}\right ) \sqrt {e} \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 )}{15 d^{3/4} \sqrt {c+d x^2}}+\frac {\sqrt [4]{c} \left (15 a^2+\frac {b c (7 b c-18 a d)}{d^2}\right ) \sqrt {e} \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 )}{15 d^{3/4} \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [C] time = 0.12, size = 111, normalized size = 0.30 \[ \frac {2 \sqrt {e x} \left (3 x \sqrt {\frac {c}{d x^2}+1} \left (15 a^2 d^2-18 a b c d+7 b^2 c^2\right ) \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-\frac {c}{d x^2}\right )+b x \left (c+d x^2\right ) \left (18 a d-7 b c+5 b d x^2\right )\right )}{45 d^2 \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {e x}}{\sqrt {d x^{2} + c}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {e x}}{\sqrt {d x^{2} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 604, normalized size = 1.61 \[ \frac {\sqrt {e x}\, \left (10 b^{2} d^{3} x^{6}+36 a b \,d^{3} x^{4}-4 b^{2} c \,d^{2} x^{4}+36 a b c \,d^{2} x^{2}-14 b^{2} c^{2} d \,x^{2}+90 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a^{2} c \,d^{2} \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-45 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a^{2} c \,d^{2} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-108 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a b \,c^{2} d \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )+54 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a b \,c^{2} d \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )+42 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, b^{2} c^{3} \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-21 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, b^{2} c^{3} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )\right )}{45 \sqrt {d \,x^{2}+c}\, d^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {e x}}{\sqrt {d x^{2} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {e\,x}\,{\left (b\,x^2+a\right )}^2}{\sqrt {d\,x^2+c}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 5.69, size = 143, normalized size = 0.38 \[ \frac {a^{2} \left (e x\right )^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} e \Gamma \left (\frac {7}{4}\right )} + \frac {a b \left (e x\right )^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt {c} e^{3} \Gamma \left (\frac {11}{4}\right )} + \frac {b^{2} \left (e x\right )^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} e^{5} \Gamma \left (\frac {15}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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