Optimal. Leaf size=406 \[ -\frac {2 \sqrt {b} c \sqrt {\frac {a x^2}{b}+1} \left (a c^2+b d^2\right ) \sqrt {\frac {a (c+d x)}{a c-\sqrt {-a} \sqrt {b} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-a} \sqrt {b} d}{a c-\sqrt {-a} \sqrt {b} d}\right )}{5 (-a)^{3/2} d x \sqrt {a+\frac {b}{x^2}} \sqrt {c+d x}}+\frac {2 \sqrt {b} \sqrt {\frac {a x^2}{b}+1} \sqrt {c+d x} \left (a c^2-3 b d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-a} \sqrt {b} d}{a c-\sqrt {-a} \sqrt {b} d}\right )}{5 (-a)^{3/2} d x \sqrt {a+\frac {b}{x^2}} \sqrt {\frac {a (c+d x)}{a c-\sqrt {-a} \sqrt {b} d}}}+\frac {2 \left (a x^2+b\right ) (c+d x)^{3/2}}{5 a x \sqrt {a+\frac {b}{x^2}}}+\frac {2 c \left (a x^2+b\right ) \sqrt {c+d x}}{5 a x \sqrt {a+\frac {b}{x^2}}} \]
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Rubi [A] time = 0.46, antiderivative size = 406, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1450, 833, 844, 719, 424, 419} \[ -\frac {2 \sqrt {b} c \sqrt {\frac {a x^2}{b}+1} \left (a c^2+b d^2\right ) \sqrt {\frac {a (c+d x)}{a c-\sqrt {-a} \sqrt {b} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-a} \sqrt {b} d}{a c-\sqrt {-a} \sqrt {b} d}\right )}{5 (-a)^{3/2} d x \sqrt {a+\frac {b}{x^2}} \sqrt {c+d x}}+\frac {2 \sqrt {b} \sqrt {\frac {a x^2}{b}+1} \sqrt {c+d x} \left (a c^2-3 b d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-a} \sqrt {b} d}{a c-\sqrt {-a} \sqrt {b} d}\right )}{5 (-a)^{3/2} d x \sqrt {a+\frac {b}{x^2}} \sqrt {\frac {a (c+d x)}{a c-\sqrt {-a} \sqrt {b} d}}}+\frac {2 \left (a x^2+b\right ) (c+d x)^{3/2}}{5 a x \sqrt {a+\frac {b}{x^2}}}+\frac {2 c \left (a x^2+b\right ) \sqrt {c+d x}}{5 a x \sqrt {a+\frac {b}{x^2}}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 719
Rule 833
Rule 844
Rule 1450
Rubi steps
\begin {align*} \int \frac {(c+d x)^{3/2}}{\sqrt {a+\frac {b}{x^2}}} \, dx &=\frac {\sqrt {b+a x^2} \int \frac {x (c+d x)^{3/2}}{\sqrt {b+a x^2}} \, dx}{\sqrt {a+\frac {b}{x^2}} x}\\ &=\frac {2 (c+d x)^{3/2} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {\left (2 \sqrt {b+a x^2}\right ) \int \frac {\left (-\frac {3 b d}{2}+\frac {3 a c x}{2}\right ) \sqrt {c+d x}}{\sqrt {b+a x^2}} \, dx}{5 a \sqrt {a+\frac {b}{x^2}} x}\\ &=\frac {2 c \sqrt {c+d x} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {2 (c+d x)^{3/2} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {\left (4 \sqrt {b+a x^2}\right ) \int \frac {-3 a b c d+\frac {3}{4} a \left (a c^2-3 b d^2\right ) x}{\sqrt {c+d x} \sqrt {b+a x^2}} \, dx}{15 a^2 \sqrt {a+\frac {b}{x^2}} x}\\ &=\frac {2 c \sqrt {c+d x} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {2 (c+d x)^{3/2} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {\left (\left (a c^2-3 b d^2\right ) \sqrt {b+a x^2}\right ) \int \frac {\sqrt {c+d x}}{\sqrt {b+a x^2}} \, dx}{5 a d \sqrt {a+\frac {b}{x^2}} x}-\frac {\left (c \left (a c^2+b d^2\right ) \sqrt {b+a x^2}\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {b+a x^2}} \, dx}{5 a d \sqrt {a+\frac {b}{x^2}} x}\\ &=\frac {2 c \sqrt {c+d x} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {2 (c+d x)^{3/2} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {\left (2 \sqrt {-a} \sqrt {b} \left (a c^2-3 b d^2\right ) \sqrt {c+d x} \sqrt {1+\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {-a} \sqrt {b} d x^2}{a c-\sqrt {-a} \sqrt {b} d}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}}}{\sqrt {2}}\right )}{5 a^2 d \sqrt {a+\frac {b}{x^2}} x \sqrt {\frac {a (c+d x)}{a c-\sqrt {-a} \sqrt {b} d}}}-\frac {\left (2 \sqrt {-a} \sqrt {b} c \left (a c^2+b d^2\right ) \sqrt {\frac {a (c+d x)}{a c-\sqrt {-a} \sqrt {b} d}} \sqrt {1+\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {-a} \sqrt {b} d x^2}{a c-\sqrt {-a} \sqrt {b} d}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}}}{\sqrt {2}}\right )}{5 a^2 d \sqrt {a+\frac {b}{x^2}} x \sqrt {c+d x}}\\ &=\frac {2 c \sqrt {c+d x} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {2 (c+d x)^{3/2} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {2 \sqrt {b} \left (a c^2-3 b d^2\right ) \sqrt {c+d x} \sqrt {1+\frac {a x^2}{b}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-a} \sqrt {b} d}{a c-\sqrt {-a} \sqrt {b} d}\right )}{5 (-a)^{3/2} d \sqrt {a+\frac {b}{x^2}} x \sqrt {\frac {a (c+d x)}{a c-\sqrt {-a} \sqrt {b} d}}}-\frac {2 \sqrt {b} c \left (a c^2+b d^2\right ) \sqrt {\frac {a (c+d x)}{a c-\sqrt {-a} \sqrt {b} d}} \sqrt {1+\frac {a x^2}{b}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-a} \sqrt {b} d}{a c-\sqrt {-a} \sqrt {b} d}\right )}{5 (-a)^{3/2} d \sqrt {a+\frac {b}{x^2}} x \sqrt {c+d x}}\\ \end {align*}
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Mathematica [C] time = 3.03, size = 540, normalized size = 1.33 \[ \frac {\sqrt {c+d x} \left (\frac {2 \left (a x^2+b\right ) (2 c+d x)}{a}+\frac {2 \left (\sqrt {a} (c+d x)^{3/2} \left (-i a^{3/2} c^3+a \sqrt {b} c^2 d+3 i \sqrt {a} b c d^2-3 b^{3/2} d^3\right ) \sqrt {\frac {d \left (x+\frac {i \sqrt {b}}{\sqrt {a}}\right )}{c+d x}} \sqrt {-\frac {-d x+\frac {i \sqrt {b} d}{\sqrt {a}}}{c+d x}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-c-\frac {i \sqrt {b} d}{\sqrt {a}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {a} c-i \sqrt {b} d}{\sqrt {a} c+i \sqrt {b} d}\right )+d^2 \sqrt {-c-\frac {i \sqrt {b} d}{\sqrt {a}}} \left (a^2 c^2 x^2+a b \left (c^2-3 d^2 x^2\right )-3 b^2 d^2\right )-\sqrt {a} \sqrt {b} d (c+d x)^{3/2} \left (4 i \sqrt {a} \sqrt {b} c d+a c^2-3 b d^2\right ) \sqrt {\frac {d \left (x+\frac {i \sqrt {b}}{\sqrt {a}}\right )}{c+d x}} \sqrt {-\frac {-d x+\frac {i \sqrt {b} d}{\sqrt {a}}}{c+d x}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-c-\frac {i \sqrt {b} d}{\sqrt {a}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {a} c-i \sqrt {b} d}{\sqrt {a} c+i \sqrt {b} d}\right )\right )}{a^2 d^2 (c+d x) \sqrt {-c-\frac {i \sqrt {b} d}{\sqrt {a}}}}\right )}{5 x \sqrt {a+\frac {b}{x^2}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d x^{3} + c x^{2}\right )} \sqrt {d x + c} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}, x\right ) \]
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 \[ \int \frac {{\left (d x + c\right )}^{\frac {3}{2}}}{\sqrt {a + \frac {b}{x^{2}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 1145, normalized size = 2.82 \[ \frac {\frac {2 a^{2} d^{4} x^{4}}{5}+\frac {6 a^{2} c \,d^{3} x^{3}}{5}+\frac {4 a^{2} c^{2} d^{2} x^{2}}{5}+\frac {2 a b \,d^{4} x^{2}}{5}-\frac {2 \sqrt {\frac {\left (-a x +\sqrt {-a b}\right ) d}{a c +\sqrt {-a b}\, d}}\, \sqrt {\frac {\left (a x +\sqrt {-a b}\right ) d}{-a c +\sqrt {-a b}\, d}}\, \sqrt {-\frac {\left (d x +c \right ) a}{-a c +\sqrt {-a b}\, d}}\, a^{2} c^{4} \EllipticE \left (\sqrt {-\frac {\left (d x +c \right ) a}{-a c +\sqrt {-a b}\, d}}, \sqrt {-\frac {-a c +\sqrt {-a b}\, d}{a c +\sqrt {-a b}\, d}}\right )}{5}+\frac {4 \sqrt {\frac {\left (-a x +\sqrt {-a b}\right ) d}{a c +\sqrt {-a b}\, d}}\, \sqrt {\frac {\left (a x +\sqrt {-a b}\right ) d}{-a c +\sqrt {-a b}\, d}}\, \sqrt {-\frac {\left (d x +c \right ) a}{-a c +\sqrt {-a b}\, d}}\, a b \,c^{2} d^{2} \EllipticE \left (\sqrt {-\frac {\left (d x +c \right ) a}{-a c +\sqrt {-a b}\, d}}, \sqrt {-\frac {-a c +\sqrt {-a b}\, d}{a c +\sqrt {-a b}\, d}}\right )}{5}-\frac {6 \sqrt {\frac {\left (-a x +\sqrt {-a b}\right ) d}{a c +\sqrt {-a b}\, d}}\, \sqrt {\frac {\left (a x +\sqrt {-a b}\right ) d}{-a c +\sqrt {-a b}\, d}}\, \sqrt {-\frac {\left (d x +c \right ) a}{-a c +\sqrt {-a b}\, d}}\, a b \,c^{2} d^{2} \EllipticF \left (\sqrt {-\frac {\left (d x +c \right ) a}{-a c +\sqrt {-a b}\, d}}, \sqrt {-\frac {-a c +\sqrt {-a b}\, d}{a c +\sqrt {-a b}\, d}}\right )}{5}+\frac {6 a b c \,d^{3} x}{5}+\frac {6 \sqrt {\frac {\left (-a x +\sqrt {-a b}\right ) d}{a c +\sqrt {-a b}\, d}}\, \sqrt {\frac {\left (a x +\sqrt {-a b}\right ) d}{-a c +\sqrt {-a b}\, d}}\, \sqrt {-\frac {\left (d x +c \right ) a}{-a c +\sqrt {-a b}\, d}}\, b^{2} d^{4} \EllipticE \left (\sqrt {-\frac {\left (d x +c \right ) a}{-a c +\sqrt {-a b}\, d}}, \sqrt {-\frac {-a c +\sqrt {-a b}\, d}{a c +\sqrt {-a b}\, d}}\right )}{5}-\frac {6 \sqrt {\frac {\left (-a x +\sqrt {-a b}\right ) d}{a c +\sqrt {-a b}\, d}}\, \sqrt {\frac {\left (a x +\sqrt {-a b}\right ) d}{-a c +\sqrt {-a b}\, d}}\, \sqrt {-\frac {\left (d x +c \right ) a}{-a c +\sqrt {-a b}\, d}}\, b^{2} d^{4} \EllipticF \left (\sqrt {-\frac {\left (d x +c \right ) a}{-a c +\sqrt {-a b}\, d}}, \sqrt {-\frac {-a c +\sqrt {-a b}\, d}{a c +\sqrt {-a b}\, d}}\right )}{5}+\frac {4 a b \,c^{2} d^{2}}{5}+\frac {2 \sqrt {\frac {\left (-a x +\sqrt {-a b}\right ) d}{a c +\sqrt {-a b}\, d}}\, \sqrt {\frac {\left (a x +\sqrt {-a b}\right ) d}{-a c +\sqrt {-a b}\, d}}\, \sqrt {-\frac {\left (d x +c \right ) a}{-a c +\sqrt {-a b}\, d}}\, \sqrt {-a b}\, a \,c^{3} d \EllipticF \left (\sqrt {-\frac {\left (d x +c \right ) a}{-a c +\sqrt {-a b}\, d}}, \sqrt {-\frac {-a c +\sqrt {-a b}\, d}{a c +\sqrt {-a b}\, d}}\right )}{5}+\frac {2 \sqrt {\frac {\left (-a x +\sqrt {-a b}\right ) d}{a c +\sqrt {-a b}\, d}}\, \sqrt {\frac {\left (a x +\sqrt {-a b}\right ) d}{-a c +\sqrt {-a b}\, d}}\, \sqrt {-\frac {\left (d x +c \right ) a}{-a c +\sqrt {-a b}\, d}}\, \sqrt {-a b}\, b c \,d^{3} \EllipticF \left (\sqrt {-\frac {\left (d x +c \right ) a}{-a c +\sqrt {-a b}\, d}}, \sqrt {-\frac {-a c +\sqrt {-a b}\, d}{a c +\sqrt {-a b}\, d}}\right )}{5}}{\sqrt {d x +c}\, \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, a^{2} d^{2} x} \]
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 \[ \int \frac {{\left (d x + c\right )}^{\frac {3}{2}}}{\sqrt {a + \frac {b}{x^{2}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c+d\,x\right )}^{3/2}}{\sqrt {a+\frac {b}{x^2}}} \,d x \]
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 \[ \int \frac {\left (c + d x\right )^{\frac {3}{2}}}{\sqrt {a + \frac {b}{x^{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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