Optimal. Leaf size=406 \[ \frac {2 c \sqrt {c+d x} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {2 (c+d x)^{3/2} \left (b+a x^2\right )}{5 a \sqrt {a+\frac {b}{x^2}} x}+\frac {2 \sqrt {b} \left (a c^2-3 b d^2\right ) \sqrt {c+d x} \sqrt {1+\frac {a x^2}{b}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-a} \sqrt {b} d}{a c-\sqrt {-a} \sqrt {b} d}\right )}{5 (-a)^{3/2} d \sqrt {a+\frac {b}{x^2}} x \sqrt {\frac {a (c+d x)}{a c-\sqrt {-a} \sqrt {b} d}}}-\frac {2 \sqrt {b} c \left (a c^2+b d^2\right ) \sqrt {\frac {a (c+d x)}{a c-\sqrt {-a} \sqrt {b} d}} \sqrt {1+\frac {a x^2}{b}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-a} \sqrt {b} d}{a c-\sqrt {-a} \sqrt {b} d}\right )}{5 (-a)^{3/2} d \sqrt {a+\frac {b}{x^2}} x \sqrt {c+d x}} \]
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Rubi [A]
time = 0.32, 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 = {1464, 847, 858,
733, 435, 430} \begin {gather*} -\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 (\text {ArcSin}\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 (\text {ArcSin}\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}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 435
Rule 733
Rule 847
Rule 858
Rule 1464
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 ) \text {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 ) \text {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] Result contains complex when optimal does not.
time = 21.94, size = 540, normalized size = 1.33 \begin {gather*} \frac {\sqrt {c+d x} \left (\frac {2 (2 c+d x) \left (b+a x^2\right )}{a}+\frac {2 \left (d^2 \sqrt {-c-\frac {i \sqrt {b} d}{\sqrt {a}}} \left (-3 b^2 d^2+a^2 c^2 x^2+a b \left (c^2-3 d^2 x^2\right )\right )+\sqrt {a} \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 (\frac {i \sqrt {b}}{\sqrt {a}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {i \sqrt {b} d}{\sqrt {a}}-d x}{c+d x}} (c+d x)^{3/2} 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 )-\sqrt {a} \sqrt {b} d \left (a c^2+4 i \sqrt {a} \sqrt {b} c d-3 b d^2\right ) \sqrt {\frac {d \left (\frac {i \sqrt {b}}{\sqrt {a}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {i \sqrt {b} d}{\sqrt {a}}-d x}{c+d x}} (c+d x)^{3/2} 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 \sqrt {-c-\frac {i \sqrt {b} d}{\sqrt {a}}} (c+d x)}\right )}{5 \sqrt {a+\frac {b}{x^2}} x} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1144\) vs.
\(2(330)=660\).
time = 0.36, size = 1145, normalized size = 2.82
method | result | size |
risch | \(\frac {2 \left (d x +2 c \right ) \left (a \,x^{2}+b \right ) \sqrt {d x +c}}{5 a \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}+\frac {\left (\frac {2 \left (c^{2} a -3 d^{2} b \right ) \left (\frac {c}{d}-\frac {\sqrt {-a b}}{a}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}\, \sqrt {\frac {x -\frac {\sqrt {-a b}}{a}}{-\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}\, \sqrt {\frac {x +\frac {\sqrt {-a b}}{a}}{-\frac {c}{d}+\frac {\sqrt {-a b}}{a}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {-a b}}{a}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {-a b}}{a}}{-\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}\right )+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {-a b}}{a}}{-\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}\right )}{a}\right )}{\sqrt {a d \,x^{3}+a c \,x^{2}+b d x +b c}}-\frac {8 b c d \left (\frac {c}{d}-\frac {\sqrt {-a b}}{a}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}\, \sqrt {\frac {x -\frac {\sqrt {-a b}}{a}}{-\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}\, \sqrt {\frac {x +\frac {\sqrt {-a b}}{a}}{-\frac {c}{d}+\frac {\sqrt {-a b}}{a}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {-a b}}{a}}{-\frac {c}{d}-\frac {\sqrt {-a b}}{a}}}\right )}{\sqrt {a d \,x^{3}+a c \,x^{2}+b d x +b c}}\right ) \sqrt {\left (a \,x^{2}+b \right ) \left (d x +c \right )}}{5 a \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x \sqrt {d x +c}}\) | \(623\) |
default | \(\frac {\frac {2 \sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}\, \sqrt {\frac {\left (-a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d +a c}}\, \sqrt {\frac {\left (a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d -a c}}\, \EllipticF \left (\sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}, \sqrt {-\frac {\sqrt {-a b}\, d -a c}{\sqrt {-a b}\, d +a c}}\right ) \sqrt {-a b}\, a \,c^{3} d}{5}+\frac {2 \sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}\, \sqrt {\frac {\left (-a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d +a c}}\, \sqrt {\frac {\left (a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d -a c}}\, \EllipticF \left (\sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}, \sqrt {-\frac {\sqrt {-a b}\, d -a c}{\sqrt {-a b}\, d +a c}}\right ) \sqrt {-a b}\, b c \,d^{3}}{5}-\frac {6 \sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}\, \sqrt {\frac {\left (-a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d +a c}}\, \sqrt {\frac {\left (a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d -a c}}\, \EllipticF \left (\sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}, \sqrt {-\frac {\sqrt {-a b}\, d -a c}{\sqrt {-a b}\, d +a c}}\right ) a b \,c^{2} d^{2}}{5}-\frac {6 \sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}\, \sqrt {\frac {\left (-a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d +a c}}\, \sqrt {\frac {\left (a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d -a c}}\, \EllipticF \left (\sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}, \sqrt {-\frac {\sqrt {-a b}\, d -a c}{\sqrt {-a b}\, d +a c}}\right ) b^{2} d^{4}}{5}-\frac {2 \sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}\, \sqrt {\frac {\left (-a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d +a c}}\, \sqrt {\frac {\left (a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d -a c}}\, \EllipticE \left (\sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}, \sqrt {-\frac {\sqrt {-a b}\, d -a c}{\sqrt {-a b}\, d +a c}}\right ) a^{2} c^{4}}{5}+\frac {4 \sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}\, \sqrt {\frac {\left (-a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d +a c}}\, \sqrt {\frac {\left (a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d -a c}}\, \EllipticE \left (\sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}, \sqrt {-\frac {\sqrt {-a b}\, d -a c}{\sqrt {-a b}\, d +a c}}\right ) a b \,c^{2} d^{2}}{5}+\frac {6 \sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}\, \sqrt {\frac {\left (-a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d +a c}}\, \sqrt {\frac {\left (a x +\sqrt {-a b}\right ) d}{\sqrt {-a b}\, d -a c}}\, \EllipticE \left (\sqrt {-\frac {\left (d x +c \right ) a}{\sqrt {-a b}\, d -a c}}, \sqrt {-\frac {\sqrt {-a b}\, d -a c}{\sqrt {-a b}\, d +a c}}\right ) b^{2} d^{4}}{5}+\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 {6 a b c \,d^{3} x}{5}+\frac {4 a b \,c^{2} d^{2}}{5}}{\sqrt {d x +c}\, d^{2} a^{2} x \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\) | \(1145\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.10, size = 235, normalized size = 0.58 \begin {gather*} -\frac {2 \, {\left ({\left (a c^{3} + 9 \, b c d^{2}\right )} \sqrt {a d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a c^{2} - 3 \, b d^{2}\right )}}{3 \, a d^{2}}, -\frac {8 \, {\left (a c^{3} + 9 \, b c d^{2}\right )}}{27 \, a d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (a c^{2} d - 3 \, b d^{3}\right )} \sqrt {a d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (a c^{2} - 3 \, b d^{2}\right )}}{3 \, a d^{2}}, -\frac {8 \, {\left (a c^{3} + 9 \, b c d^{2}\right )}}{27 \, a d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a c^{2} - 3 \, b d^{2}\right )}}{3 \, a d^{2}}, -\frac {8 \, {\left (a c^{3} + 9 \, b c d^{2}\right )}}{27 \, a d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (a d^{3} x^{2} + 2 \, a c d^{2} x\right )} \sqrt {d x + c} \sqrt {\frac {a x^{2} + b}{x^{2}}}\right )}}{15 \, a^{2} d^{2}} \end {gather*}
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 \begin {gather*} \int \frac {\left (c + d x\right )^{\frac {3}{2}}}{\sqrt {a + \frac {b}{x^{2}}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{3/2}}{\sqrt {a+\frac {b}{x^2}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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