Optimal. Leaf size=410 \[ \frac {16 d \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{9 \sqrt [4]{3} \sqrt {a+b x} (b c-a d)^{7/3} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {16 d \sqrt [6]{c+d x}}{9 \sqrt {a+b x} (b c-a d)^2}-\frac {2 \sqrt [6]{c+d x}}{3 (a+b x)^{3/2} (b c-a d)} \]
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Rubi [A] time = 0.27, antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {51, 63, 225} \[ \frac {16 d \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{9 \sqrt [4]{3} \sqrt {a+b x} (b c-a d)^{7/3} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {16 d \sqrt [6]{c+d x}}{9 \sqrt {a+b x} (b c-a d)^2}-\frac {2 \sqrt [6]{c+d x}}{3 (a+b x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 225
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{5/2} (c+d x)^{5/6}} \, dx &=-\frac {2 \sqrt [6]{c+d x}}{3 (b c-a d) (a+b x)^{3/2}}-\frac {(8 d) \int \frac {1}{(a+b x)^{3/2} (c+d x)^{5/6}} \, dx}{9 (b c-a d)}\\ &=-\frac {2 \sqrt [6]{c+d x}}{3 (b c-a d) (a+b x)^{3/2}}+\frac {16 d \sqrt [6]{c+d x}}{9 (b c-a d)^2 \sqrt {a+b x}}+\frac {\left (16 d^2\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/6}} \, dx}{27 (b c-a d)^2}\\ &=-\frac {2 \sqrt [6]{c+d x}}{3 (b c-a d) (a+b x)^{3/2}}+\frac {16 d \sqrt [6]{c+d x}}{9 (b c-a d)^2 \sqrt {a+b x}}+\frac {(32 d) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^6}{d}}} \, dx,x,\sqrt [6]{c+d x}\right )}{9 (b c-a d)^2}\\ &=-\frac {2 \sqrt [6]{c+d x}}{3 (b c-a d) (a+b x)^{3/2}}+\frac {16 d \sqrt [6]{c+d x}}{9 (b c-a d)^2 \sqrt {a+b x}}+\frac {16 d \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{9 \sqrt [4]{3} (b c-a d)^{7/3} \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 73, normalized size = 0.18 \[ -\frac {2 \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \, _2F_1\left (-\frac {3}{2},\frac {5}{6};-\frac {1}{2};\frac {d (a+b x)}{a d-b c}\right )}{3 b (a+b x)^{3/2} (c+d x)^{5/6}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} {\left (d x + c\right )}^{\frac {1}{6}}}{b^{3} d x^{4} + a^{3} c + {\left (b^{3} c + 3 \, a b^{2} d\right )} x^{3} + 3 \, {\left (a b^{2} c + a^{2} b d\right )} x^{2} + {\left (3 \, a^{2} b c + a^{3} d\right )} x}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {5}{2}} {\left (d x + c\right )}^{\frac {5}{6}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b x +a \right )^{\frac {5}{2}} \left (d x +c \right )^{\frac {5}{6}}}\, dx \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {5}{2}} {\left (d x + c\right )}^{\frac {5}{6}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/6}} \,d x \]
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 \[ \int \frac {1}{\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{6}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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