Optimal. Leaf size=344 \[ -\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {8 b d^2 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}+\frac {8 b d \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {4 b \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {c (d+e x)}{c d+e}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.60, antiderivative size = 344, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {43, 5247, 12, 6721, 6742, 719, 424, 944, 419, 932, 168, 538, 537} \[ -\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {8 b d^2 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}+\frac {8 b d \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {4 b \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {c (d+e x)}{c d+e}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 43
Rule 168
Rule 419
Rule 424
Rule 537
Rule 538
Rule 719
Rule 932
Rule 944
Rule 5247
Rule 6721
Rule 6742
Rubi steps
\begin {align*} \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx &=-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {b \int \frac {2 (-2 d+e x) \sqrt {d+e x}}{3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{c}\\ &=-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {(2 b) \int \frac {(-2 d+e x) \sqrt {d+e x}}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{3 c e^2}\\ &=-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {(-2 d+e x) \sqrt {d+e x}}{x \sqrt {1-c^2 x^2}} \, dx}{3 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \left (\frac {e \sqrt {d+e x}}{\sqrt {1-c^2 x^2}}-\frac {2 d \sqrt {d+e x}}{x \sqrt {1-c^2 x^2}}\right ) \, dx}{3 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {\left (4 b d \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{x \sqrt {1-c^2 x^2}} \, dx}{3 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{3 c e \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {\left (4 b d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{3 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b d \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{3 c e \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{3 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}}\\ &=-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {4 b \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 )}{3 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {\left (4 b d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \sqrt {d+e x}} \, dx}{3 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (8 b d \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{3 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {4 b \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 )}{3 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {8 b d \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 )}{3 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (8 b d^2 \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 )}{3 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {4 b \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 )}{3 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {8 b d \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 )}{3 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (8 b d^2 \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 )}{3 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {4 b \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 )}{3 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {8 b d \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 )}{3 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {8 b d^2 \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 )}{3 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 1.51, size = 289, normalized size = 0.84 \[ \frac {2 \left (a \sqrt {d+e x} (e x-2 d)+\frac {2 i b \sqrt {\frac {e (c x+1)}{e-c d}} \sqrt {\frac {e-c e x}{c d+e}} \left ((c d+e) F\left (i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right )|\frac {c d+e}{c d-e}\right )+(c d-e) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right )|\frac {c d+e}{c d-e}\right )-2 c d \Pi \left (\frac {e}{c d}+1;i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right )|\frac {c d+e}{c d-e}\right )\right )}{c^2 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {-\frac {c}{c d+e}}}+b \csc ^{-1}(c x) \sqrt {d+e x} (e x-2 d)\right )}{3 e^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x}{\sqrt {e x + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 412, normalized size = 1.20 \[ \frac {2 a \left (\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}-d \sqrt {e x +d}\right )+2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \mathrm {arccsc}\left (c x \right )}{3}-\mathrm {arccsc}\left (c x \right ) d \sqrt {e x +d}-\frac {2 \left (d \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \sqrt {\frac {d c -e}{d c +e}}\right ) c +\EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \sqrt {\frac {d c -e}{d c +e}}\right ) c d -2 d \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \frac {d c -e}{c d}, \frac {\sqrt {\frac {c}{d c +e}}}{\sqrt {\frac {c}{d c -e}}}\right ) c -\EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \sqrt {\frac {d c -e}{d c +e}}\right ) e +\EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \sqrt {\frac {d c -e}{d c +e}}\right ) e \right ) \sqrt {-\frac {\left (e x +d \right ) c -d c -e}{d c +e}}\, \sqrt {-\frac {\left (e x +d \right ) c -d c +e}{d c -e}}}{3 c^{2} \sqrt {\frac {c}{d c -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {d+e\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________