Optimal. Leaf size=315 \[ \frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}-\frac {4 b \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b d^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.34, antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 11, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {5335, 1588,
972, 733, 430, 947, 174, 552, 551, 858, 435} \begin {gather*} \frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}-\frac {4 b d^2 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {4 b d \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 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 (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {c (d+e x)}{c d+e}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 174
Rule 430
Rule 435
Rule 551
Rule 552
Rule 733
Rule 858
Rule 947
Rule 972
Rule 1588
Rule 5335
Rubi steps
\begin {align*} \int \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {(2 b) \int \frac {(d+e x)^{3/2}}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{3 c e}\\ &=\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {(d+e x)^{3/2}}{x \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{3 c e \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \left (\frac {2 d e}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}}+\frac {d^2}{x \sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}}+\frac {e^2 x}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}}\right ) \, dx}{3 c e \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {\left (4 b d \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{3 c \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b d^2 \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{3 c e \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b e \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {x}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{3 c \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {-\frac {1}{c^2}+x^2}} \, dx}{3 c \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (2 b d \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{3 c \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 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 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (8 b d \sqrt {\frac {d+e x}{d+\frac {e}{c}}} \sqrt {1-c^2 x^2}\right ) \text {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 )}{3 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 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 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (4 b d^2 \sqrt {1-c^2 x^2}\right ) \text {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 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \text {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 )}{3 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {d+e x}{d+\frac {e}{c}}}}+\frac {\left (4 b d \sqrt {\frac {d+e x}{d+\frac {e}{c}}} \sqrt {1-c^2 x^2}\right ) \text {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 )}{3 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}-\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 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 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 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (4 b d^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2}\right ) \text {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 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}-\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 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 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 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 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 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(657\) vs. \(2(315)=630\).
time = 28.38, size = 657, normalized size = 2.09 \begin {gather*} \frac {2 a (d+e x)^{3/2}}{3 e}-\frac {b (c d+c e x) \left (-\frac {2 \left (2 e \sqrt {1-\frac {1}{c^2 x^2}}+c d \csc ^{-1}(c x)+c e x \csc ^{-1}(c x)\right )}{e}+\frac {4 d \sqrt {-c^2 \left (1-\frac {1}{c^2 x^2}\right ) x^2} F\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{(c d+e) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c d+c e x}{c d+e}}}-\frac {4 (-c d+e) \sqrt {-c^2 \left (1-\frac {1}{c^2 x^2}\right ) x^2} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c d+c e x}{c d+e}}}+\frac {\left (c^2 \left (1-\frac {1}{c^2 x^2}\right ) x^2 (c d+c e x)+c^2 d x \sqrt {-c^2 \left (1-\frac {1}{c^2 x^2}\right ) x^2} \sqrt {\frac {c d+c e x}{c d+e}} F\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )-\frac {c x (1+c x) \sqrt {\frac {e-c e x}{c d+e}} \sqrt {\frac {c d+c e x}{c d-e}} \left ((c d+e) E\left (\text {ArcSin}\left (\sqrt {\frac {c d+c e x}{c d-e}}\right )|\frac {c d-e}{c d+e}\right )-e F\left (\text {ArcSin}\left (\sqrt {\frac {c d+c e x}{c d-e}}\right )|\frac {c d-e}{c d+e}\right )\right )}{\sqrt {\frac {e (1+c x)}{-c d+e}}}+c e x \sqrt {-c^2 \left (1-\frac {1}{c^2 x^2}\right ) x^2} \sqrt {\frac {c d+c e x}{c d+e}} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )\right ) \sec \left (2 \csc ^{-1}(c x)\right ) \sin \left (4 \csc ^{-1}(c x)\right )}{c^2 \left (1-\frac {1}{c^2 x^2}\right ) \left (e+\frac {d}{x}\right ) x^2}\right )}{3 c^2 \sqrt {d+e x}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.57, size = 386, normalized size = 1.23
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (e x +d \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \mathrm {arccsc}\left (c x \right )}{3}+\frac {2 \left (2 d \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c -\EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d -d \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c +\EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e -\EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e \right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}}{3 c^{2} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e}\) | \(386\) |
default | \(\frac {\frac {2 \left (e x +d \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \mathrm {arccsc}\left (c x \right )}{3}+\frac {2 \left (2 d \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c -\EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d -d \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c +\EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e -\EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e \right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}}{3 c^{2} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e}\) | \(386\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {acsc}{\left (c x \right )}\right ) \sqrt {d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )\,\sqrt {d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________