Optimal. Leaf size=333 \[ \frac {4 a^{7/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (7 \sqrt {a} B+15 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {8 a^{9/4} B \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {8 a^2 B x \sqrt {a+c x^2}}{15 \sqrt {c} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {4 a \sqrt {e x} \sqrt {a+c x^2} (15 A+7 B x)}{105 e}+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (9 A+7 B x)}{63 e} \]
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Rubi [A] time = 0.35, antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {815, 842, 840, 1198, 220, 1196} \[ \frac {4 a^{7/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (7 \sqrt {a} B+15 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {8 a^{9/4} B \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {8 a^2 B x \sqrt {a+c x^2}}{15 \sqrt {c} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {4 a \sqrt {e x} \sqrt {a+c x^2} (15 A+7 B x)}{105 e}+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (9 A+7 B x)}{63 e} \]
Antiderivative was successfully verified.
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Rule 220
Rule 815
Rule 840
Rule 842
Rule 1196
Rule 1198
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{\sqrt {e x}} \, dx &=\frac {2 \sqrt {e x} (9 A+7 B x) \left (a+c x^2\right )^{3/2}}{63 e}+\frac {4 \int \frac {\left (\frac {9}{2} a A c e^2+\frac {7}{2} a B c e^2 x\right ) \sqrt {a+c x^2}}{\sqrt {e x}} \, dx}{21 c e^2}\\ &=\frac {4 a \sqrt {e x} (15 A+7 B x) \sqrt {a+c x^2}}{105 e}+\frac {2 \sqrt {e x} (9 A+7 B x) \left (a+c x^2\right )^{3/2}}{63 e}+\frac {16 \int \frac {\frac {45}{4} a^2 A c^2 e^4+\frac {21}{4} a^2 B c^2 e^4 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{315 c^2 e^4}\\ &=\frac {4 a \sqrt {e x} (15 A+7 B x) \sqrt {a+c x^2}}{105 e}+\frac {2 \sqrt {e x} (9 A+7 B x) \left (a+c x^2\right )^{3/2}}{63 e}+\frac {\left (16 \sqrt {x}\right ) \int \frac {\frac {45}{4} a^2 A c^2 e^4+\frac {21}{4} a^2 B c^2 e^4 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{315 c^2 e^4 \sqrt {e x}}\\ &=\frac {4 a \sqrt {e x} (15 A+7 B x) \sqrt {a+c x^2}}{105 e}+\frac {2 \sqrt {e x} (9 A+7 B x) \left (a+c x^2\right )^{3/2}}{63 e}+\frac {\left (32 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {\frac {45}{4} a^2 A c^2 e^4+\frac {21}{4} a^2 B c^2 e^4 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{315 c^2 e^4 \sqrt {e x}}\\ &=\frac {4 a \sqrt {e x} (15 A+7 B x) \sqrt {a+c x^2}}{105 e}+\frac {2 \sqrt {e x} (9 A+7 B x) \left (a+c x^2\right )^{3/2}}{63 e}-\frac {\left (8 a^{5/2} B \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{15 \sqrt {c} \sqrt {e x}}+\frac {\left (8 a^2 \left (7 \sqrt {a} B+15 A \sqrt {c}\right ) \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{105 \sqrt {c} \sqrt {e x}}\\ &=\frac {4 a \sqrt {e x} (15 A+7 B x) \sqrt {a+c x^2}}{105 e}+\frac {8 a^2 B x \sqrt {a+c x^2}}{15 \sqrt {c} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {2 \sqrt {e x} (9 A+7 B x) \left (a+c x^2\right )^{3/2}}{63 e}-\frac {8 a^{9/4} B \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {4 a^{7/4} \left (7 \sqrt {a} B+15 A \sqrt {c}\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 83, normalized size = 0.25 \[ \frac {2 a x \sqrt {a+c x^2} \left (3 A \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {c x^2}{a}\right )+B x \, _2F_1\left (-\frac {3}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^2}{a}\right )\right )}{3 \sqrt {e x} \sqrt {\frac {c x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B c x^{3} + A c x^{2} + B a x + A a\right )} \sqrt {c x^{2} + a} \sqrt {e x}}{e x}, x\right ) \]
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 \[ \int \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{\sqrt {e x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 338, normalized size = 1.02 \[ \frac {\frac {2 B \,c^{3} x^{6}}{9}+\frac {2 A \,c^{3} x^{5}}{7}+\frac {32 B a \,c^{2} x^{4}}{45}+\frac {8 A a \,c^{2} x^{3}}{7}+\frac {22 B \,a^{2} c \,x^{2}}{45}+\frac {6 A \,a^{2} c x}{7}+\frac {8 \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, B \,a^{3} \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{15}-\frac {4 \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, B \,a^{3} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{15}+\frac {4 \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-a c}\, A \,a^{2} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{7}}{\sqrt {c \,x^{2}+a}\, \sqrt {e x}\, c} \]
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 \[ \int \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{\sqrt {e x}}\,{d x} \]
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 \[ \int \frac {{\left (c\,x^2+a\right )}^{3/2}\,\left (A+B\,x\right )}{\sqrt {e\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 8.35, size = 199, normalized size = 0.60 \[ \frac {A a^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {5}{4}\right )} + \frac {A \sqrt {a} c x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {9}{4}\right )} + \frac {B a^{\frac {3}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {7}{4}\right )} + \frac {B \sqrt {a} c x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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