Optimal. Leaf size=366 \[ \frac {4 \sqrt {-a} c^{3/2} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 e^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {4 c d \sqrt {a+c x^2}}{3 e \sqrt {d+e x} \left (a e^2+c d^2\right )}-\frac {4 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 e^2 \sqrt {a+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a+c x^2}}{3 e (d+e x)^{3/2}} \]
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Rubi [A] time = 0.27, antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {733, 835, 844, 719, 424, 419} \[ \frac {4 \sqrt {-a} c^{3/2} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 e^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {4 c d \sqrt {a+c x^2}}{3 e \sqrt {d+e x} \left (a e^2+c d^2\right )}-\frac {4 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 e^2 \sqrt {a+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a+c x^2}}{3 e (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 719
Rule 733
Rule 835
Rule 844
Rubi steps
\begin {align*} \int \frac {\sqrt {a+c x^2}}{(d+e x)^{5/2}} \, dx &=-\frac {2 \sqrt {a+c x^2}}{3 e (d+e x)^{3/2}}+\frac {(2 c) \int \frac {x}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx}{3 e}\\ &=-\frac {2 \sqrt {a+c x^2}}{3 e (d+e x)^{3/2}}+\frac {4 c d \sqrt {a+c x^2}}{3 e \left (c d^2+a e^2\right ) \sqrt {d+e x}}-\frac {(4 c) \int \frac {-\frac {a e}{2}+\frac {c d x}{2}}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 e \left (c d^2+a e^2\right )}\\ &=-\frac {2 \sqrt {a+c x^2}}{3 e (d+e x)^{3/2}}+\frac {4 c d \sqrt {a+c x^2}}{3 e \left (c d^2+a e^2\right ) \sqrt {d+e x}}+\frac {(2 c) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 e^2}-\frac {\left (2 c^2 d\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{3 e^2 \left (c d^2+a e^2\right )}\\ &=-\frac {2 \sqrt {a+c x^2}}{3 e (d+e x)^{3/2}}+\frac {4 c d \sqrt {a+c x^2}}{3 e \left (c d^2+a e^2\right ) \sqrt {d+e x}}-\frac {\left (4 a c^{3/2} d \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^2 \left (c d^2+a e^2\right ) \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (4 a \sqrt {c} \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} e^2 \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=-\frac {2 \sqrt {a+c x^2}}{3 e (d+e x)^{3/2}}+\frac {4 c d \sqrt {a+c x^2}}{3 e \left (c d^2+a e^2\right ) \sqrt {d+e x}}+\frac {4 \sqrt {-a} c^{3/2} d \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^2 \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {4 \sqrt {-a} \sqrt {c} \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 e^2 \sqrt {d+e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 1.09, size = 504, normalized size = 1.38 \[ \frac {2 \sqrt {a+c x^2} \left (c d (d+2 e x)-a e^2\right )}{3 (d+e x)^{3/2} \left (a e^3+c d^2 e\right )}-\frac {4 c \left (d e^2 \left (a+c x^2\right ) \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}-\sqrt {a} e (d+e x)^{3/2} \left (\sqrt {c} d+i \sqrt {a} e\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {c} d (d+e x)^{3/2} \left (\sqrt {a} e-i \sqrt {c} d\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{3 e^3 \sqrt {a+c x^2} \sqrt {d+e x} \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} \sqrt {e x + d}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]
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 \[ \int \frac {\sqrt {c x^{2} + a}}{{\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 1309, normalized size = 3.58 \[ \text {result too large to display} \]
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 \[ \int \frac {\sqrt {c x^{2} + a}}{{\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {c\,x^2+a}}{{\left (d+e\,x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\sqrt {a + c x^{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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