Optimal. Leaf size=352 \[ -\frac {4 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \left (a B e^2-3 A c d e+4 B c d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {c} e^3 \sqrt {a+c x^2} \sqrt {d+e x}}+\frac {4 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} (4 B d-3 A e) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 e^3 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {2 \sqrt {a+c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}} \]
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Rubi [A] time = 0.25, antiderivative size = 352, 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 = {813, 844, 719, 424, 419} \[ -\frac {4 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \left (a B e^2-3 A c d e+4 B c d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {c} e^3 \sqrt {a+c x^2} \sqrt {d+e x}}+\frac {2 \sqrt {a+c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}+\frac {4 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} (4 B d-3 A e) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 e^3 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 719
Rule 813
Rule 844
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {a+c x^2}}{(d+e x)^{3/2}} \, dx &=\frac {2 (4 B d-3 A e+B e x) \sqrt {a+c x^2}}{3 e^2 \sqrt {d+e x}}-\frac {2 \int \frac {-a B e+c (4 B d-3 A e) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 e^2}\\ &=\frac {2 (4 B d-3 A e+B e x) \sqrt {a+c x^2}}{3 e^2 \sqrt {d+e x}}-\frac {(2 c (4 B d-3 A e)) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{3 e^3}+\frac {\left (2 \left (4 B c d^2-3 A c d e+a B e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 e^3}\\ &=\frac {2 (4 B d-3 A e+B e x) \sqrt {a+c x^2}}{3 e^2 \sqrt {d+e x}}-\frac {\left (4 a \sqrt {c} (4 B d-3 A 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 )}{3 \sqrt {-a} e^3 \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (4 a \left (4 B c d^2-3 A c d e+a B 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 )}{3 \sqrt {-a} \sqrt {c} e^3 \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=\frac {2 (4 B d-3 A e+B e x) \sqrt {a+c x^2}}{3 e^2 \sqrt {d+e x}}+\frac {4 \sqrt {-a} \sqrt {c} (4 B d-3 A 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 )}{3 e^3 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {4 \sqrt {-a} \left (4 B c d^2-3 A c d e+a B 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 )}{3 \sqrt {c} e^3 \sqrt {d+e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 3.28, size = 512, normalized size = 1.45 \[ \frac {\sqrt {d+e x} \left (\frac {2 \left (a+c x^2\right ) (-3 A e+4 B d+B e x)}{e^2 (d+e x)}+\frac {2 (d+e x) \left (\frac {2 e^2 \left (a+c x^2\right ) \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (3 A e-4 B d)}{(d+e x)^2}-\frac {2 \sqrt {a} e \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} \left (-i \sqrt {a} B e+3 A \sqrt {c} e-4 B \sqrt {c} d\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {d+e x}}+\frac {2 \sqrt {c} \left (\sqrt {a} e-i \sqrt {c} d\right ) (3 A e-4 B d) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {d+e x}}\right )}{e^4 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{3 \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.13, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} {\left (B x + A\right )} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]
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 \[ \int \frac {\sqrt {c x^{2} + a} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 1465, normalized size = 4.16 \[ \text {result too large to display} \]
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 \[ \int \frac {\sqrt {c x^{2} + a} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {c\,x^2+a}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B x\right ) \sqrt {a + c x^{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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