Optimal. Leaf size=345 \[ -\frac {\sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} (A c d-3 a B e) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {-a} c^{3/2} \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {\sqrt {d+e x} (a (A e+B d)-x (A c d-a B e))}{a c \sqrt {a+c x^2}}+\frac {A \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {-a} c^{3/2} \sqrt {a+c x^2} \sqrt {d+e x}} \]
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Rubi [A] time = 0.29, antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {819, 844, 719, 424, 419} \[ -\frac {\sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} (A c d-3 a B e) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {-a} c^{3/2} \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {\sqrt {d+e x} (a (A e+B d)-x (A c d-a B e))}{a c \sqrt {a+c x^2}}+\frac {A \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {-a} c^{3/2} \sqrt {a+c x^2} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 719
Rule 819
Rule 844
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a+c x^2\right )^{3/2}} \, dx &=-\frac {\sqrt {d+e x} (a (B d+A e)-(A c d-a B e) x)}{a c \sqrt {a+c x^2}}+\frac {\int \frac {\frac {1}{2} a e (3 B d+A e)-\frac {1}{2} e (A c d-3 a B e) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{a c}\\ &=-\frac {\sqrt {d+e x} (a (B d+A e)-(A c d-a B e) x)}{a c \sqrt {a+c x^2}}+\frac {(-A c d+3 a B e) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{2 a c}+\frac {\left (A \left (c d^2+a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{2 a c}\\ &=-\frac {\sqrt {d+e x} (a (B d+A e)-(A c d-a B e) x)}{a c \sqrt {a+c x^2}}+\frac {\left ((-A c d+3 a B e) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{\sqrt {-a} c^{3/2} \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (A \left (c d^2+a e^2\right ) \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{\sqrt {-a} c^{3/2} \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=-\frac {\sqrt {d+e x} (a (B d+A e)-(A c d-a B e) x)}{a c \sqrt {a+c x^2}}-\frac {(A c d-3 a B e) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {-a} c^{3/2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {A \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {-a} c^{3/2} \sqrt {d+e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 4.80, size = 465, normalized size = 1.35 \[ \frac {\sqrt {d+e x} \left (\frac {e \left (a+c x^2\right ) (3 a B e-A c d)}{d+e x}+c (A c d x-a (A e+B (d+e x)))-\sqrt {a} c \sqrt {d+e x} \left (A \sqrt {c}+3 i \sqrt {a} B\right ) \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )-\frac {i c \sqrt {d+e x} \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} (A c d-3 a B e) E\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{e}\right )}{a c^2 \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B e x^{2} + A d + {\left (B d + A e\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}}{c^{2} x^{4} + 2 \, a c x^{2} + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 1319, normalized size = 3.82 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{3/2}}{{\left (c\,x^2+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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