Optimal. Leaf size=220 \[ \frac {99 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {1432 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{5/2}}-\frac {4437 \sqrt {1-2 x} \sqrt {3+5 x}}{12005 (2+3 x)^{3/2}}-\frac {27618 \sqrt {1-2 x} \sqrt {3+5 x}}{84035 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac {9206 \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{84035}-\frac {7738 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{84035} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {100, 155, 157,
164, 114, 120} \begin {gather*} -\frac {7738 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{84035}+\frac {9206 \sqrt {33} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{84035}+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{5/2}}-\frac {27618 \sqrt {1-2 x} \sqrt {5 x+3}}{84035 \sqrt {3 x+2}}-\frac {4437 \sqrt {1-2 x} \sqrt {5 x+3}}{12005 (3 x+2)^{3/2}}-\frac {1432 \sqrt {1-2 x} \sqrt {5 x+3}}{1715 (3 x+2)^{5/2}}+\frac {99 \sqrt {5 x+3}}{49 \sqrt {1-2 x} (3 x+2)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 100
Rule 114
Rule 120
Rule 155
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{7/2}} \, dx &=\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac {1}{21} \int \frac {\left (-\frac {147}{2}-150 x\right ) \sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{7/2}} \, dx\\ &=\frac {99 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{5/2}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac {1}{147} \int \frac {-4959-\frac {17025 x}{2}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx\\ &=\frac {99 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {1432 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{5/2}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac {2 \int \frac {-\frac {72609}{4}-32220 x}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{5145}\\ &=\frac {99 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {1432 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{5/2}}-\frac {4437 \sqrt {1-2 x} \sqrt {3+5 x}}{12005 (2+3 x)^{3/2}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac {4 \int \frac {-\frac {91683}{4}-\frac {199665 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{108045}\\ &=\frac {99 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {1432 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{5/2}}-\frac {4437 \sqrt {1-2 x} \sqrt {3+5 x}}{12005 (2+3 x)^{3/2}}-\frac {27618 \sqrt {1-2 x} \sqrt {3+5 x}}{84035 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac {8 \int \frac {\frac {362655}{8}+\frac {621405 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{756315}\\ &=\frac {99 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {1432 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{5/2}}-\frac {4437 \sqrt {1-2 x} \sqrt {3+5 x}}{12005 (2+3 x)^{3/2}}-\frac {27618 \sqrt {1-2 x} \sqrt {3+5 x}}{84035 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac {27618 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{84035}+\frac {42559 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{84035}\\ &=\frac {99 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {1432 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{5/2}}-\frac {4437 \sqrt {1-2 x} \sqrt {3+5 x}}{12005 (2+3 x)^{3/2}}-\frac {27618 \sqrt {1-2 x} \sqrt {3+5 x}}{84035 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac {9206 \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{84035}-\frac {7738 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{84035}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 8.96, size = 110, normalized size = 0.50 \begin {gather*} \frac {-\frac {2 \sqrt {3+5 x} \left (-88623-640441 x-718167 x^2+1056186 x^3+1491372 x^4\right )}{(1-2 x)^{3/2} (2+3 x)^{5/2}}+3 \sqrt {2} \left (-9206 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+51765 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{252105} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(397\) vs.
\(2(162)=324\).
time = 0.10, size = 398, normalized size = 1.81
method | result | size |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {121 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{7203 \left (-\frac {1}{2}+x \right )^{2}}-\frac {2596 \left (-30 x^{2}-38 x -12\right )}{50421 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{46305 \left (\frac {2}{3}+x \right )^{3}}-\frac {14638 \left (-30 x^{2}-3 x +9\right )}{252105 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {34 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{15435 \left (\frac {2}{3}+x \right )^{2}}-\frac {8059 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{352947 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {9206 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{117649 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(301\) |
default | \(-\frac {\sqrt {1-2 x}\, \left (2298186 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+497124 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1915155 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+414270 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-510708 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-110472 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-510708 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-110472 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+14913720 x^{5}+19510092 x^{4}-844554 x^{3}-10713412 x^{2}-4728876 x -531738\right )}{252105 \left (2+3 x \right )^{\frac {5}{2}} \left (-1+2 x \right )^{2} \sqrt {3+5 x}}\) | \(398\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.26, size = 70, normalized size = 0.32 \begin {gather*} -\frac {2 \, {\left (1491372 \, x^{4} + 1056186 \, x^{3} - 718167 \, x^{2} - 640441 \, x - 88623\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{252105 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________