Optimal. Leaf size=189 \[ -\frac {3 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d^2}+\frac {(b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{32 b^2 d}+\frac {(b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b}+\frac {3 (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{5/2} d^{5/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps
used = 7, 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)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{5/2} d^{5/2}}-\frac {3 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^3}{64 b^2 d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x} (b c-a d)^2}{32 b^2 d}+\frac {(a+b x)^{5/2} \sqrt {c+d x} (b c-a d)}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps
\begin {align*} \int (a+b x)^{3/2} (c+d x)^{3/2} \, dx &=\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b}+\frac {(3 (b c-a d)) \int (a+b x)^{3/2} \sqrt {c+d x} \, dx}{8 b}\\ &=\frac {(b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b}+\frac {(b c-a d)^2 \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{16 b^2}\\ &=\frac {(b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{32 b^2 d}+\frac {(b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b}-\frac {\left (3 (b c-a d)^3\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{64 b^2 d}\\ &=-\frac {3 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d^2}+\frac {(b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{32 b^2 d}+\frac {(b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b}+\frac {\left (3 (b c-a d)^4\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 b^2 d^2}\\ &=-\frac {3 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d^2}+\frac {(b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{32 b^2 d}+\frac {(b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b}+\frac {\left (3 (b c-a d)^4\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 )}{64 b^3 d^2}\\ &=-\frac {3 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d^2}+\frac {(b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{32 b^2 d}+\frac {(b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b}+\frac {\left (3 (b c-a d)^4\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 )}{64 b^3 d^2}\\ &=-\frac {3 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d^2}+\frac {(b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{32 b^2 d}+\frac {(b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b}+\frac {3 (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{5/2} d^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 144, normalized size = 0.76 \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 d^3 (a+b x)^3-11 b d^2 (a+b x)^2 (c+d x)-11 b^2 d (a+b x) (c+d x)^2+3 b^3 (c+d x)^3\right )}{64 b^2 d^2}+\frac {3 (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{64 b^{5/2} 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 = 0.16, size = 206, normalized size = 1.09
method | result | size |
default | \(\frac {\left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {5}{2}}}{4 d}-\frac {3 \left (-a d +b c \right ) \left (\frac {\sqrt {b x +a}\, \left (d x +c \right )^{\frac {5}{2}}}{3 d}-\frac {\left (-a d +b c \right ) \left (\frac {\left (d x +c \right )^{\frac {3}{2}} \sqrt {b x +a}}{2 b}-\frac {3 \left (a d -b c \right ) \left (\frac {\sqrt {b x +a}\, \sqrt {d x +c}}{b}-\frac {\left (a d -b c \right ) \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d x}{\sqrt {b d}}+\sqrt {b d \,x^{2}+\left (a d +b c \right ) x +a c}\right )}{2 b \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}}\right )}{4 b}\right )}{6 d}\right )}{8 d}\) | \(206\) |
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 = 0.33, size = 534, normalized size = 2.83 \begin {gather*} \left [\frac {3 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\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 (16 \, b^{4} d^{4} x^{3} - 3 \, b^{4} c^{3} d + 11 \, a b^{3} c^{2} d^{2} + 11 \, a^{2} b^{2} c d^{3} - 3 \, a^{3} b d^{4} + 24 \, {\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (b^{4} c^{2} d^{2} + 22 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{256 \, b^{3} d^{3}}, -\frac {3 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\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 (16 \, b^{4} d^{4} x^{3} - 3 \, b^{4} c^{3} d + 11 \, a b^{3} c^{2} d^{2} + 11 \, a^{2} b^{2} c d^{3} - 3 \, a^{3} b d^{4} + 24 \, {\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (b^{4} c^{2} d^{2} + 22 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{128 \, b^{3} 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 \left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1071 vs.
\(2 (151) = 302\).
time = 0.11, size = 1443, normalized size = 7.63
result too large to display
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 {\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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