Optimal. Leaf size=113 \[ -\frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}+\frac {3 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 \sqrt {b} d^{5/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {52, 65, 223,
212} \begin {gather*} \frac {3 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 \sqrt {b} d^{5/2}}-\frac {3 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}{4 d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx &=\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {(3 (b c-a d)) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{4 d}\\ &=-\frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}+\frac {\left (3 (b c-a d)^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 d^2}\\ &=-\frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}+\frac {\left (3 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{4 b d^2}\\ &=-\frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}+\frac {\left (3 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 b d^2}\\ &=-\frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}+\frac {3 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 \sqrt {b} d^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 94, normalized size = 0.83 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} (-3 b c+5 a d+2 b d x)}{4 d^2}+\frac {3 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{4 \sqrt {b} d^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.15, size = 140, normalized size = 1.24
method | result | size |
default | \(\frac {\left (b x +a \right )^{\frac {3}{2}} \sqrt {d x +c}}{2 d}-\frac {3 \left (-a d +b c \right ) \left (\frac {\sqrt {b x +a}\, \sqrt {d x +c}}{d}-\frac {\left (-a d +b c \right ) \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d x}{\sqrt {b d}}+\sqrt {b d \,x^{2}+\left (a d +b c \right ) x +a c}\right )}{2 d \sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {b d}}\right )}{4 d}\) | \(140\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 306, normalized size = 2.71 \begin {gather*} \left [\frac {3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (2 \, b^{2} d^{2} x - 3 \, b^{2} c d + 5 \, a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, b d^{3}}, -\frac {3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (2 \, b^{2} d^{2} x - 3 \, b^{2} c d + 5 \, a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, b d^{3}}\right ] \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 (a + b x\right )^{\frac {3}{2}}}{\sqrt {c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.01, size = 186, normalized size = 1.65 \begin {gather*} \frac {b^{2} \left (2 \left (\frac {\frac {1}{8}\cdot 2 d^{2} \sqrt {a+b x} \sqrt {a+b x}}{b d^{3}}-\frac {\frac {1}{8} \left (3 b d c-3 d^{2} a\right )}{b d^{3}}\right ) \sqrt {a+b x} \sqrt {-a b d+b^{2} c+b d \left (a+b x\right )}+\frac {2 \left (-3 a^{2} d^{2}+6 a b c d-3 b^{2} c^{2}\right ) \ln \left |\sqrt {-a b d+b^{2} c+b d \left (a+b x\right )}-\sqrt {b d} \sqrt {a+b x}\right |}{8 d^{2} \sqrt {b d}}\right )}{\left |b\right | b} \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 = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{3/2}}{\sqrt {c+d\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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