Optimal. Leaf size=441 \[ -\frac {2 \sqrt {c+d x^2} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right )}{195 c^2 x^{5/2}}+\frac {2 d^{5/4} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{195 c^{11/4} \sqrt {c+d x^2}}-\frac {4 d^{5/4} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{195 c^{11/4} \sqrt {c+d x^2}}-\frac {4 d \sqrt {c+d x^2} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right )}{195 c^3 \sqrt {x}}+\frac {4 d^{3/2} \sqrt {x} \sqrt {c+d x^2} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right )}{195 c^3 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{13 c x^{13/2}}-\frac {2 a \left (c+d x^2\right )^{3/2} (26 b c-7 a d)}{117 c^2 x^{9/2}} \]
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Rubi [A] time = 0.39, antiderivative size = 437, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {462, 453, 277, 325, 329, 305, 220, 1196} \[ -\frac {4 d \sqrt {c+d x^2} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right )}{195 c^3 \sqrt {x}}+\frac {4 d^{3/2} \sqrt {x} \sqrt {c+d x^2} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right )}{195 c^3 \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {2 d^{5/4} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{195 c^{11/4} \sqrt {c+d x^2}}-\frac {4 d^{5/4} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{195 c^{11/4} \sqrt {c+d x^2}}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{13 c x^{13/2}}-\frac {2 \sqrt {c+d x^2} \left (39 b^2-\frac {a d (26 b c-7 a d)}{c^2}\right )}{195 x^{5/2}}-\frac {2 a \left (c+d x^2\right )^{3/2} (26 b c-7 a d)}{117 c^2 x^{9/2}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 277
Rule 305
Rule 325
Rule 329
Rule 453
Rule 462
Rule 1196
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{15/2}} \, dx &=-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{13 c x^{13/2}}+\frac {2 \int \frac {\left (\frac {1}{2} a (26 b c-7 a d)+\frac {13}{2} b^2 c x^2\right ) \sqrt {c+d x^2}}{x^{11/2}} \, dx}{13 c}\\ &=-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{13 c x^{13/2}}-\frac {2 a (26 b c-7 a d) \left (c+d x^2\right )^{3/2}}{117 c^2 x^{9/2}}-\frac {1}{39} \left (-39 b^2+\frac {a d (26 b c-7 a d)}{c^2}\right ) \int \frac {\sqrt {c+d x^2}}{x^{7/2}} \, dx\\ &=-\frac {2 \left (39 b^2-\frac {a d (26 b c-7 a d)}{c^2}\right ) \sqrt {c+d x^2}}{195 x^{5/2}}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{13 c x^{13/2}}-\frac {2 a (26 b c-7 a d) \left (c+d x^2\right )^{3/2}}{117 c^2 x^{9/2}}+\frac {1}{195} \left (2 d \left (39 b^2-\frac {a d (26 b c-7 a d)}{c^2}\right )\right ) \int \frac {1}{x^{3/2} \sqrt {c+d x^2}} \, dx\\ &=-\frac {2 \left (39 b^2-\frac {a d (26 b c-7 a d)}{c^2}\right ) \sqrt {c+d x^2}}{195 x^{5/2}}-\frac {4 d \left (39 b^2 c^2-26 a b c d+7 a^2 d^2\right ) \sqrt {c+d x^2}}{195 c^3 \sqrt {x}}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{13 c x^{13/2}}-\frac {2 a (26 b c-7 a d) \left (c+d x^2\right )^{3/2}}{117 c^2 x^{9/2}}+\frac {\left (2 d^2 \left (39 b^2 c^2-26 a b c d+7 a^2 d^2\right )\right ) \int \frac {\sqrt {x}}{\sqrt {c+d x^2}} \, dx}{195 c^3}\\ &=-\frac {2 \left (39 b^2-\frac {a d (26 b c-7 a d)}{c^2}\right ) \sqrt {c+d x^2}}{195 x^{5/2}}-\frac {4 d \left (39 b^2 c^2-26 a b c d+7 a^2 d^2\right ) \sqrt {c+d x^2}}{195 c^3 \sqrt {x}}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{13 c x^{13/2}}-\frac {2 a (26 b c-7 a d) \left (c+d x^2\right )^{3/2}}{117 c^2 x^{9/2}}+\frac {\left (4 d^2 \left (39 b^2 c^2-26 a b c d+7 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c+d x^4}} \, dx,x,\sqrt {x}\right )}{195 c^3}\\ &=-\frac {2 \left (39 b^2-\frac {a d (26 b c-7 a d)}{c^2}\right ) \sqrt {c+d x^2}}{195 x^{5/2}}-\frac {4 d \left (39 b^2 c^2-26 a b c d+7 a^2 d^2\right ) \sqrt {c+d x^2}}{195 c^3 \sqrt {x}}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{13 c x^{13/2}}-\frac {2 a (26 b c-7 a d) \left (c+d x^2\right )^{3/2}}{117 c^2 x^{9/2}}+\frac {\left (4 d^{3/2} \left (39 b^2 c^2-26 a b c d+7 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+d x^4}} \, dx,x,\sqrt {x}\right )}{195 c^{5/2}}-\frac {\left (4 d^{3/2} \left (39 b^2 c^2-26 a b c d+7 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {d} x^2}{\sqrt {c}}}{\sqrt {c+d x^4}} \, dx,x,\sqrt {x}\right )}{195 c^{5/2}}\\ &=-\frac {2 \left (39 b^2-\frac {a d (26 b c-7 a d)}{c^2}\right ) \sqrt {c+d x^2}}{195 x^{5/2}}-\frac {4 d \left (39 b^2 c^2-26 a b c d+7 a^2 d^2\right ) \sqrt {c+d x^2}}{195 c^3 \sqrt {x}}+\frac {4 d^{3/2} \left (39 b^2 c^2-26 a b c d+7 a^2 d^2\right ) \sqrt {x} \sqrt {c+d x^2}}{195 c^3 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{13 c x^{13/2}}-\frac {2 a (26 b c-7 a d) \left (c+d x^2\right )^{3/2}}{117 c^2 x^{9/2}}-\frac {4 d^{5/4} \left (39 b^2 c^2-26 a b c d+7 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{195 c^{11/4} \sqrt {c+d x^2}}+\frac {2 d^{5/4} \left (39 b^2 c^2-26 a b c d+7 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{195 c^{11/4} \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [C] time = 0.20, size = 182, normalized size = 0.41 \[ \frac {4 d^2 x^8 \sqrt {\frac {d x^2}{c}+1} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right ) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {d x^2}{c}\right )-2 \left (c+d x^2\right ) \left (a^2 \left (45 c^3+10 c^2 d x^2-14 c d^2 x^4+42 d^3 x^6\right )+26 a b c x^2 \left (5 c^2+2 c d x^2-6 d^2 x^4\right )+117 b^2 c^2 x^4 \left (c+2 d x^2\right )\right )}{585 c^3 x^{13/2} \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {d x^{2} + c}}{x^{\frac {15}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}}{x^{\frac {15}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 706, normalized size = 1.60 \[ \frac {-\frac {28 a^{2} d^{4} x^{8}}{195}+\frac {8 a b c \,d^{3} x^{8}}{15}-\frac {4 b^{2} c^{2} d^{2} x^{8}}{5}+\frac {28 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a^{2} c \,d^{3} x^{6} \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{195}-\frac {14 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a^{2} c \,d^{3} x^{6} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{195}-\frac {8 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a b \,c^{2} d^{2} x^{6} \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{15}+\frac {4 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a b \,c^{2} d^{2} x^{6} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{15}+\frac {4 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, b^{2} c^{3} d \,x^{6} \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{5}-\frac {2 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, b^{2} c^{3} d \,x^{6} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{5}-\frac {56 a^{2} c \,d^{3} x^{6}}{585}+\frac {16 a b \,c^{2} d^{2} x^{6}}{45}-\frac {6 b^{2} c^{3} d \,x^{6}}{5}+\frac {8 a^{2} c^{2} d^{2} x^{4}}{585}-\frac {28 a b \,c^{3} d \,x^{4}}{45}-\frac {2 b^{2} c^{4} x^{4}}{5}-\frac {22 a^{2} c^{3} d \,x^{2}}{117}-\frac {4 a b \,c^{4} x^{2}}{9}-\frac {2 a^{2} c^{4}}{13}}{\sqrt {d \,x^{2}+c}\, c^{3} x^{\frac {13}{2}}} \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}}{x^{\frac {15}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c}}{x^{15/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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