Optimal. Leaf size=341 \[ \frac {4 a^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (5 \sqrt {a} B+21 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{35 \sqrt [4]{c} e \sqrt {e x} \sqrt {a+c x^2}}-\frac {24 a^{5/4} A \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 e \sqrt {e x} \sqrt {a+c x^2}}+\frac {4 \sqrt {e x} \sqrt {a+c x^2} (5 a B+21 A c x)}{35 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (7 A-B x)}{7 e \sqrt {e x}}+\frac {24 a A \sqrt {c} x \sqrt {a+c x^2}}{5 e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )} \]
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Rubi [A] time = 0.35, antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {813, 815, 842, 840, 1198, 220, 1196} \[ \frac {4 a^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (5 \sqrt {a} B+21 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{35 \sqrt [4]{c} e \sqrt {e x} \sqrt {a+c x^2}}-\frac {24 a^{5/4} A \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 e \sqrt {e x} \sqrt {a+c x^2}}+\frac {4 \sqrt {e x} \sqrt {a+c x^2} (5 a B+21 A c x)}{35 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (7 A-B x)}{7 e \sqrt {e x}}+\frac {24 a A \sqrt {c} x \sqrt {a+c x^2}}{5 e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )} \]
Antiderivative was successfully verified.
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Rule 220
Rule 813
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}}{(e x)^{3/2}} \, dx &=-\frac {2 (7 A-B x) \left (a+c x^2\right )^{3/2}}{7 e \sqrt {e x}}-\frac {6 \int \frac {(-a B e-7 A c e x) \sqrt {a+c x^2}}{\sqrt {e x}} \, dx}{7 e^2}\\ &=\frac {4 \sqrt {e x} (5 a B+21 A c x) \sqrt {a+c x^2}}{35 e^2}-\frac {2 (7 A-B x) \left (a+c x^2\right )^{3/2}}{7 e \sqrt {e x}}-\frac {8 \int \frac {-\frac {5}{2} a^2 B c e^3-\frac {21}{2} a A c^2 e^3 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{35 c e^4}\\ &=\frac {4 \sqrt {e x} (5 a B+21 A c x) \sqrt {a+c x^2}}{35 e^2}-\frac {2 (7 A-B x) \left (a+c x^2\right )^{3/2}}{7 e \sqrt {e x}}-\frac {\left (8 \sqrt {x}\right ) \int \frac {-\frac {5}{2} a^2 B c e^3-\frac {21}{2} a A c^2 e^3 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{35 c e^4 \sqrt {e x}}\\ &=\frac {4 \sqrt {e x} (5 a B+21 A c x) \sqrt {a+c x^2}}{35 e^2}-\frac {2 (7 A-B x) \left (a+c x^2\right )^{3/2}}{7 e \sqrt {e x}}-\frac {\left (16 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {-\frac {5}{2} a^2 B c e^3-\frac {21}{2} a A c^2 e^3 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{35 c e^4 \sqrt {e x}}\\ &=\frac {4 \sqrt {e x} (5 a B+21 A c x) \sqrt {a+c x^2}}{35 e^2}-\frac {2 (7 A-B x) \left (a+c x^2\right )^{3/2}}{7 e \sqrt {e x}}+\frac {\left (8 a^{3/2} \left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{35 e \sqrt {e x}}-\frac {\left (24 a^{3/2} A \sqrt {c} \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 )}{5 e \sqrt {e x}}\\ &=\frac {24 a A \sqrt {c} x \sqrt {a+c x^2}}{5 e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {4 \sqrt {e x} (5 a B+21 A c x) \sqrt {a+c x^2}}{35 e^2}-\frac {2 (7 A-B x) \left (a+c x^2\right )^{3/2}}{7 e \sqrt {e x}}-\frac {24 a^{5/4} A \sqrt [4]{c} \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 )}{5 e \sqrt {e x} \sqrt {a+c x^2}}+\frac {4 a^{5/4} \left (5 \sqrt {a} B+21 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 )}{35 \sqrt [4]{c} e \sqrt {e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 81, normalized size = 0.24 \[ \frac {2 a x \sqrt {a+c x^2} \left (B x \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {c x^2}{a}\right )-A \, _2F_1\left (-\frac {3}{2},-\frac {1}{4};\frac {3}{4};-\frac {c x^2}{a}\right )\right )}{(e x)^{3/2} \sqrt {\frac {c x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B c x^{3} + A c x^{2} + B a x + A a\right )} \sqrt {c x^{2} + a} \sqrt {e x}}{e^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{\left (e x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 340, normalized size = 1.00 \[ -\frac {2 \left (-5 B \,c^{3} x^{5}-7 A \,c^{3} x^{4}-20 B a \,c^{2} x^{3}+28 A a \,c^{2} x^{2}-84 \sqrt {2}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, A \,a^{2} c \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )+42 \sqrt {2}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, A \,a^{2} c \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )-15 B \,a^{2} c x +35 A \,a^{2} c -10 \sqrt {-a c}\, \sqrt {2}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, B \,a^{2} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )\right )}{35 \sqrt {c \,x^{2}+a}\, \sqrt {e x}\, c e} \]
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 \[ \int \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{\left (e x\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c\,x^2+a\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (e\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 8.62, size = 202, normalized size = 0.59 \[ \frac {A a^{\frac {3}{2}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {A \sqrt {a} c x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {B a^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {B \sqrt {a} c x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {3}{2}} \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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