Optimal. Leaf size=398 \[ -\frac {4 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \left (3 c d^2-5 a e^2\right ) \left (a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{105 c^{3/2} e^2 \sqrt {a+c x^2} \sqrt {d+e x}}+\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (-5 a e^2+3 c d^2+24 c d e x\right )}{105 c e}+\frac {4 \sqrt {-a} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (3 c d^2-29 a e^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{105 \sqrt {c} e^2 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {2 e \left (a+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c} \]
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Rubi [A] time = 0.44, antiderivative size = 398, 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 = {743, 815, 844, 719, 424, 419} \[ -\frac {4 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \left (3 c d^2-5 a e^2\right ) \left (a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{105 c^{3/2} e^2 \sqrt {a+c x^2} \sqrt {d+e x}}+\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (-5 a e^2+3 c d^2+24 c d e x\right )}{105 c e}+\frac {4 \sqrt {-a} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (3 c d^2-29 a e^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{105 \sqrt {c} e^2 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {2 e \left (a+c x^2\right )^{3/2} \sqrt {d+e x}}{7 c} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 719
Rule 743
Rule 815
Rule 844
Rubi steps
\begin {align*} \int (d+e x)^{3/2} \sqrt {a+c x^2} \, dx &=\frac {2 e \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{7 c}+\frac {2 \int \frac {\left (\frac {1}{2} \left (7 c d^2-a e^2\right )+4 c d e x\right ) \sqrt {a+c x^2}}{\sqrt {d+e x}} \, dx}{7 c}\\ &=\frac {2 \sqrt {d+e x} \left (3 c d^2-5 a e^2+24 c d e x\right ) \sqrt {a+c x^2}}{105 c e}+\frac {2 e \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{7 c}+\frac {8 \int \frac {\frac {1}{4} a c e^2 \left (27 c d^2-5 a e^2\right )-\frac {1}{4} c^2 d e \left (3 c d^2-29 a e^2\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{105 c^2 e^2}\\ &=\frac {2 \sqrt {d+e x} \left (3 c d^2-5 a e^2+24 c d e x\right ) \sqrt {a+c x^2}}{105 c e}+\frac {2 e \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{7 c}+\frac {1}{105} \left (2 d \left (29 a-\frac {3 c d^2}{e^2}\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx+\frac {\left (2 \left (3 c d^2-5 a e^2\right ) \left (c d^2+a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{105 c e^2}\\ &=\frac {2 \sqrt {d+e x} \left (3 c d^2-5 a e^2+24 c d e x\right ) \sqrt {a+c x^2}}{105 c e}+\frac {2 e \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{7 c}+\frac {\left (4 a d \left (29 a-\frac {3 c d^2}{e^2}\right ) \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 )}{105 \sqrt {-a} \sqrt {c} \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 (3 c d^2-5 a e^2\right ) \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 )}{105 \sqrt {-a} c^{3/2} e^2 \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=\frac {2 \sqrt {d+e x} \left (3 c d^2-5 a e^2+24 c d e x\right ) \sqrt {a+c x^2}}{105 c e}+\frac {2 e \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{7 c}-\frac {4 \sqrt {-a} d \left (29 a-\frac {3 c d^2}{e^2}\right ) \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 )}{105 \sqrt {c} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {4 \sqrt {-a} \left (3 c d^2-5 a e^2\right ) \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 )}{105 c^{3/2} e^2 \sqrt {d+e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 3.44, size = 582, normalized size = 1.46 \[ \frac {\sqrt {d+e x} \left (\frac {2 \left (a+c x^2\right ) \left (10 a e^2+3 c \left (d^2+8 d e x+5 e^2 x^2\right )\right )}{c e}+\frac {4 \left (\sqrt {a} e (d+e x)^{3/2} \left (-5 i a^{3/2} e^3+27 i \sqrt {a} c d^2 e-29 a \sqrt {c} d e^2+3 c^{3/2} d^3\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {c} d (d+e x)^{3/2} \left (29 a^{3/2} e^3-3 \sqrt {a} c d^2 e-29 i a \sqrt {c} d e^2+3 i c^{3/2} d^3\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )-d e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (-29 a^2 e^2+a c \left (3 d^2-29 e^2 x^2\right )+3 c^2 d^2 x^2\right )\right )}{c e^3 (d+e x) \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{105 \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.07, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c x^{2} + a} {\left (e x + d\right )}^{\frac {3}{2}}, x\right ) \]
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 \[ \int \sqrt {c x^{2} + a} {\left (e x + d\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.30, size = 1386, normalized size = 3.48 \[ \text {result too large to display} \]
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 \[ \int \sqrt {c x^{2} + a} {\left (e x + d\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int \sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^{3/2} \,d x \]
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 \[ \int \sqrt {a + c x^{2}} \left (d + e x\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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