Optimal. Leaf size=372 \[ \frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}+\frac {4 b d^3 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}+\frac {28 b d \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^2 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \sqrt {1-c^2 x^2} \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^4 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}+\frac {4 b e \left (1-c^2 x^2\right ) \sqrt {d+e x}}{15 c^3 x \sqrt {1-\frac {1}{c^2 x^2}}} \]
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Rubi [A] time = 0.76, antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 13, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {5226, 1574, 958, 719, 419, 933, 168, 538, 537, 844, 424, 931, 1584} \[ \frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}+\frac {4 b \sqrt {1-c^2 x^2} \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^4 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}+\frac {4 b d^3 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}+\frac {4 b e \left (1-c^2 x^2\right ) \sqrt {d+e x}}{15 c^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {28 b d \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^2 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {c (d+e x)}{c d+e}}} \]
Antiderivative was successfully verified.
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Rule 168
Rule 419
Rule 424
Rule 537
Rule 538
Rule 719
Rule 844
Rule 931
Rule 933
Rule 958
Rule 1574
Rule 1584
Rule 5226
Rubi steps
\begin {align*} \int (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx &=\frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {(2 b) \int \frac {(d+e x)^{5/2}}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{5 c e}\\ &=\frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {(d+e x)^{5/2}}{x \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=\frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \left (\frac {3 d^2 e}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}}+\frac {d^3}{x \sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}}+\frac {3 d e^2 x}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}}+\frac {e^3 x^2}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}}\right ) \, dx}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=\frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {\left (6 b d^2 \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (2 b d^3 \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (6 b d e \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {x}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (2 b e^2 \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {x^2}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=\frac {4 b e \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}-\frac {\left (6 b d \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (6 b d^2 \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b e \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {-\frac {e x}{c^2}+2 d x^2}{x \sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{15 c \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (2 b d^3 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \sqrt {d+e x}} \, dx}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (12 b d^2 \sqrt {\frac {d+e x}{d+\frac {e}{c}}} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1-\frac {2 e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{5 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {4 b e \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}+\frac {12 b d^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (2 b e \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {-\frac {e}{c^2}+2 d x}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{15 c \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (4 b d^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {d+\frac {e}{c}-\frac {e x^2}{c}}} \, dx,x,\sqrt {1-c x}\right )}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (12 b d \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {2 e x^2}{c \left (d+\frac {e}{c}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{5 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {d+e x}{d+\frac {e}{c}}}}-\frac {\left (12 b d^2 \sqrt {\frac {d+e x}{d+\frac {e}{c}}} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1-\frac {2 e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{5 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {4 b e \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}+\frac {12 b d \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {\left (4 b d \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {-\frac {1}{c^2}+x^2}} \, dx}{15 c \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \left (-2 d^2-\frac {e^2}{c^2}\right ) \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{15 c \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (4 b d^3 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {1-\frac {e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\sqrt {1-c x}\right )}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {4 b e \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}+\frac {12 b d \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b d^3 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (8 b d \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {2 e x^2}{c \left (d+\frac {e}{c}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{15 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {d+e x}{d+\frac {e}{c}}}}-\frac {\left (4 b \left (-2 d^2-\frac {e^2}{c^2}\right ) \sqrt {\frac {d+e x}{d+\frac {e}{c}}} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1-\frac {2 e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{15 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {4 b e \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 (d+e x)^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e}+\frac {28 b d \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^4 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b d^3 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ \end {align*}
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Mathematica [C] time = 1.48, size = 333, normalized size = 0.90 \[ \frac {1}{15} \left (\frac {6 a (d+e x)^{5/2}}{e}-\frac {4 b e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}{c}+\frac {4 i b \sqrt {\frac {e (c x+1)}{e-c d}} \sqrt {\frac {e-c e x}{c d+e}} \left (\left (9 c^2 d^2-7 c d e+e^2\right ) F\left (i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right )|\frac {c d+e}{c d-e}\right )-3 c^2 d^2 \Pi \left (\frac {e}{c d}+1;i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right )|\frac {c d+e}{c d-e}\right )-7 c d (c d-e) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right )|\frac {c d+e}{c d-e}\right )\right )}{c^3 e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {-\frac {c}{c d+e}}}+\frac {6 b \sec ^{-1}(c x) (d+e x)^{5/2}}{e}\right ) \]
Antiderivative was successfully verified.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 810, normalized size = 2.18 \[ \frac {\frac {2 \left (e x +d \right )^{\frac {5}{2}} a}{5}+2 b \left (\frac {\left (e x +d \right )^{\frac {5}{2}} \mathrm {arcsec}\left (c x \right )}{5}-\frac {2 \left (\sqrt {\frac {c}{d c -e}}\, \left (e x +d \right )^{\frac {5}{2}} c^{2}+9 d^{2} \sqrt {-\frac {\left (e x +d \right ) c -d c +e}{d c -e}}\, \sqrt {-\frac {\left (e x +d \right ) c -d c -e}{d c +e}}\, \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \sqrt {\frac {d c -e}{d c +e}}\right ) c^{2}-7 \sqrt {-\frac {\left (e x +d \right ) c -d c +e}{d c -e}}\, \sqrt {-\frac {\left (e x +d \right ) c -d c -e}{d c +e}}\, \EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \sqrt {\frac {d c -e}{d c +e}}\right ) c^{2} d^{2}-3 d^{2} \sqrt {-\frac {\left (e x +d \right ) c -d c +e}{d c -e}}\, \sqrt {-\frac {\left (e x +d \right ) c -d c -e}{d c +e}}\, \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \frac {d c -e}{c d}, \frac {\sqrt {\frac {c}{d c +e}}}{\sqrt {\frac {c}{d c -e}}}\right ) c^{2}-2 \sqrt {\frac {c}{d c -e}}\, \left (e x +d \right )^{\frac {3}{2}} c^{2} d +7 \sqrt {-\frac {\left (e x +d \right ) c -d c +e}{d c -e}}\, \sqrt {-\frac {\left (e x +d \right ) c -d c -e}{d c +e}}\, \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \sqrt {\frac {d c -e}{d c +e}}\right ) c d e -7 \sqrt {-\frac {\left (e x +d \right ) c -d c +e}{d c -e}}\, \sqrt {-\frac {\left (e x +d \right ) c -d c -e}{d c +e}}\, \EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \sqrt {\frac {d c -e}{d c +e}}\right ) c d e +\sqrt {\frac {c}{d c -e}}\, \sqrt {e x +d}\, c^{2} d^{2}+\sqrt {-\frac {\left (e x +d \right ) c -d c +e}{d c -e}}\, \sqrt {-\frac {\left (e x +d \right ) c -d c -e}{d c +e}}\, \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \sqrt {\frac {d c -e}{d c +e}}\right ) e^{2}-\sqrt {\frac {c}{d c -e}}\, \sqrt {e x +d}\, e^{2}\right )}{15 c^{3} \sqrt {\frac {c}{d c -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )\,{\left (d+e\,x\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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