Optimal. Leaf size=195 \[ \frac {2 \sqrt {\frac {3}{143}} (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {\frac {11}{23}} \sqrt {5 x+7}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{31 \sqrt {2 x-5} \sqrt {4 x+1}}+\frac {10 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {5-2 x}{5 x+7}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{22}} \sqrt {4 x+1}}{\sqrt {5 x+7}}\right )|\frac {62}{39}\right )}{713 \sqrt {2 x-5} \sqrt {\frac {2-3 x}{5 x+7}}} \]
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Rubi [A] time = 0.18, antiderivative size = 270, normalized size of antiderivative = 1.38, number of steps used = 8, number of rules used = 7, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {171, 170, 418, 176, 422, 492, 411} \[ -\frac {10 \sqrt {2 x-5} \sqrt {4 x+1}}{897 \sqrt {2-3 x} \sqrt {5 x+7}}+\frac {6 \sqrt {5 x+7} F\left (\tan ^{-1}\left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right )|-\frac {39}{23}\right )}{31 \sqrt {253} \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}-\frac {5 \sqrt {\frac {22}{31}} \sqrt {4 x+1} F\left (\tan ^{-1}\left (\frac {\sqrt {\frac {31}{11}} \sqrt {2 x-5}}{\sqrt {5 x+7}}\right )|\frac {39}{62}\right )}{1209 \sqrt {2-3 x} \sqrt {-\frac {4 x+1}{2-3 x}}}+\frac {10 \sqrt {\frac {22}{31}} \sqrt {4 x+1} E\left (\tan ^{-1}\left (\frac {\sqrt {\frac {31}{11}} \sqrt {2 x-5}}{\sqrt {5 x+7}}\right )|\frac {39}{62}\right )}{897 \sqrt {2-3 x} \sqrt {-\frac {4 x+1}{2-3 x}}} \]
Antiderivative was successfully verified.
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Rule 170
Rule 171
Rule 176
Rule 411
Rule 418
Rule 422
Rule 492
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx &=\frac {3}{31} \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx+\frac {5}{31} \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx\\ &=\frac {\left (5 \sqrt {2} \sqrt {2-3 x} \sqrt {\frac {1+4 x}{7+5 x}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {31 x^2}{11}}}{\sqrt {1+\frac {23 x^2}{22}}} \, dx,x,\frac {\sqrt {-5+2 x}}{\sqrt {7+5 x}}\right )}{1209 \sqrt {1+4 x} \sqrt {-\frac {2-3 x}{7+5 x}}}+\frac {\left (3 \sqrt {\frac {2}{253}} \sqrt {-\frac {-5+2 x}{2-3 x}} \sqrt {7+5 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{2}} \sqrt {1+\frac {31 x^2}{23}}} \, dx,x,\frac {\sqrt {1+4 x}}{\sqrt {2-3 x}}\right )}{31 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{2-3 x}}}\\ &=\frac {6 \sqrt {7+5 x} F\left (\tan ^{-1}\left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right )|-\frac {39}{23}\right )}{31 \sqrt {253} \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}+\frac {\left (5 \sqrt {2} \sqrt {2-3 x} \sqrt {\frac {1+4 x}{7+5 x}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {23 x^2}{22}} \sqrt {1+\frac {31 x^2}{11}}} \, dx,x,\frac {\sqrt {-5+2 x}}{\sqrt {7+5 x}}\right )}{1209 \sqrt {1+4 x} \sqrt {-\frac {2-3 x}{7+5 x}}}+\frac {\left (5 \sqrt {2} \sqrt {2-3 x} \sqrt {\frac {1+4 x}{7+5 x}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {23 x^2}{22}} \sqrt {1+\frac {31 x^2}{11}}} \, dx,x,\frac {\sqrt {-5+2 x}}{\sqrt {7+5 x}}\right )}{429 \sqrt {1+4 x} \sqrt {-\frac {2-3 x}{7+5 x}}}\\ &=-\frac {10 \sqrt {-5+2 x} \sqrt {1+4 x}}{897 \sqrt {2-3 x} \sqrt {7+5 x}}+\frac {6 \sqrt {7+5 x} F\left (\tan ^{-1}\left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right )|-\frac {39}{23}\right )}{31 \sqrt {253} \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}-\frac {5 \sqrt {\frac {22}{31}} \sqrt {1+4 x} F\left (\tan ^{-1}\left (\frac {\sqrt {\frac {31}{11}} \sqrt {-5+2 x}}{\sqrt {7+5 x}}\right )|\frac {39}{62}\right )}{1209 \sqrt {2-3 x} \sqrt {-\frac {1+4 x}{2-3 x}}}-\frac {\left (10 \sqrt {2} \sqrt {2-3 x} \sqrt {\frac {1+4 x}{7+5 x}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {23 x^2}{22}}}{\left (1+\frac {31 x^2}{11}\right )^{3/2}} \, dx,x,\frac {\sqrt {-5+2 x}}{\sqrt {7+5 x}}\right )}{897 \sqrt {1+4 x} \sqrt {-\frac {2-3 x}{7+5 x}}}\\ &=-\frac {10 \sqrt {-5+2 x} \sqrt {1+4 x}}{897 \sqrt {2-3 x} \sqrt {7+5 x}}+\frac {10 \sqrt {\frac {22}{31}} \sqrt {1+4 x} E\left (\tan ^{-1}\left (\frac {\sqrt {\frac {31}{11}} \sqrt {-5+2 x}}{\sqrt {7+5 x}}\right )|\frac {39}{62}\right )}{897 \sqrt {2-3 x} \sqrt {-\frac {1+4 x}{2-3 x}}}+\frac {6 \sqrt {7+5 x} F\left (\tan ^{-1}\left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right )|-\frac {39}{23}\right )}{31 \sqrt {253} \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}-\frac {5 \sqrt {\frac {22}{31}} \sqrt {1+4 x} F\left (\tan ^{-1}\left (\frac {\sqrt {\frac {31}{11}} \sqrt {-5+2 x}}{\sqrt {7+5 x}}\right )|\frac {39}{62}\right )}{1209 \sqrt {2-3 x} \sqrt {-\frac {1+4 x}{2-3 x}}}\\ \end {align*}
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Mathematica [A] time = 1.69, size = 237, normalized size = 1.22 \[ -\frac {2 \sqrt {2 x-5} \sqrt {4 x+1} \left (-23 \sqrt {682} \sqrt {\frac {8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {31}{39}} \sqrt {\frac {2 x-5}{3 x-2}}\right ),\frac {39}{62}\right )+1705 \sqrt {\frac {5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )-55 \sqrt {682} \sqrt {\frac {8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) E\left (\sin ^{-1}\left (\sqrt {\frac {31}{39}} \sqrt {\frac {2 x-5}{3 x-2}}\right )|\frac {39}{62}\right )\right )}{305877 \sqrt {2-3 x} \sqrt {5 x+7} \sqrt {\frac {5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{600 \, x^{5} - 70 \, x^{4} - 3199 \, x^{3} - 1710 \, x^{2} + 1729 \, x + 490}, x\right ) \]
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 \[ \int \frac {1}{{\left (5 \, x + 7\right )}^{\frac {3}{2}} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 599, normalized size = 3.07 \[ \frac {2 \sqrt {5 x +7}\, \sqrt {-3 x +2}\, \sqrt {2 x -5}\, \sqrt {4 x +1}\, \left (880 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, x^{2} \EllipticE \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )+1104 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, x^{2} \EllipticF \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )+7590 x^{2}+440 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, x \EllipticE \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )+552 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, x \EllipticF \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )-24035 x +55 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, \EllipticE \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )+69 \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}\, \sqrt {3}\, \sqrt {13}\, \sqrt {\frac {2 x -5}{4 x +1}}\, \sqrt {\frac {3 x -2}{4 x +1}}\, \EllipticF \left (\frac {\sqrt {31}\, \sqrt {11}\, \sqrt {\frac {5 x +7}{4 x +1}}}{31}, \frac {\sqrt {31}\, \sqrt {78}}{39}\right )+12650\right )}{305877 \left (120 x^{4}-182 x^{3}-385 x^{2}+197 x +70\right )} \]
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 \[ \int \frac {1}{{\left (5 \, x + 7\right )}^{\frac {3}{2}} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}\,{\left (5\,x+7\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1} \left (5 x + 7\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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