Optimal. Leaf size=224 \[ -\frac {3 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^2}+\frac {(b c-a d)^3 (a+b x)^{3/2} \sqrt {c+d x}}{64 b^3 d}+\frac {(b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}{16 b^3}+\frac {(b c-a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b}+\frac {3 (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{7/2} d^{5/2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps
used = 8, 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)^5 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{7/2} d^{5/2}}-\frac {3 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^4}{128 b^3 d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x} (b c-a d)^3}{64 b^3 d}+\frac {(a+b x)^{5/2} \sqrt {c+d x} (b c-a d)^2}{16 b^3}+\frac {(a+b x)^{5/2} (c+d x)^{3/2} (b c-a d)}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 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)^{5/2} \, dx &=\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b}+\frac {(b c-a d) \int (a+b x)^{3/2} (c+d x)^{3/2} \, dx}{2 b}\\ &=\frac {(b c-a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b}+\frac {\left (3 (b c-a d)^2\right ) \int (a+b x)^{3/2} \sqrt {c+d x} \, dx}{16 b^2}\\ &=\frac {(b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}{16 b^3}+\frac {(b c-a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b}+\frac {(b c-a d)^3 \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{32 b^3}\\ &=\frac {(b c-a d)^3 (a+b x)^{3/2} \sqrt {c+d x}}{64 b^3 d}+\frac {(b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}{16 b^3}+\frac {(b c-a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b}-\frac {\left (3 (b c-a d)^4\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{128 b^3 d}\\ &=-\frac {3 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^2}+\frac {(b c-a d)^3 (a+b x)^{3/2} \sqrt {c+d x}}{64 b^3 d}+\frac {(b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}{16 b^3}+\frac {(b c-a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b}+\frac {\left (3 (b c-a d)^5\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 b^3 d^2}\\ &=-\frac {3 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^2}+\frac {(b c-a d)^3 (a+b x)^{3/2} \sqrt {c+d x}}{64 b^3 d}+\frac {(b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}{16 b^3}+\frac {(b c-a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b}+\frac {\left (3 (b c-a d)^5\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 )}{128 b^4 d^2}\\ &=-\frac {3 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^2}+\frac {(b c-a d)^3 (a+b x)^{3/2} \sqrt {c+d x}}{64 b^3 d}+\frac {(b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}{16 b^3}+\frac {(b c-a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b}+\frac {\left (3 (b c-a d)^5\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 )}{128 b^4 d^2}\\ &=-\frac {3 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}{128 b^3 d^2}+\frac {(b c-a d)^3 (a+b x)^{3/2} \sqrt {c+d x}}{64 b^3 d}+\frac {(b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}{16 b^3}+\frac {(b c-a d) (a+b x)^{5/2} (c+d x)^{3/2}}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b}+\frac {3 (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{7/2} d^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.53, size = 169, normalized size = 0.75 \begin {gather*} \frac {\sqrt {b} \sqrt {d} \sqrt {a+b x} \sqrt {c+d x} \left (15 d^4 (a+b x)^4-70 b d^3 (a+b x)^3 (c+d x)+128 b^2 d^2 (a+b x)^2 (c+d x)^2+70 b^3 d (a+b x) (c+d x)^3-15 b^4 (c+d x)^4\right )+15 (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{640 b^{7/2} d^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 239, normalized size = 1.07
method | result | size |
default | \(\frac {\left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {7}{2}}}{5 d}-\frac {3 \left (-a d +b c \right ) \left (\frac {\sqrt {b x +a}\, \left (d x +c \right )^{\frac {7}{2}}}{4 d}-\frac {\left (-a d +b c \right ) \left (\frac {\left (d x +c \right )^{\frac {5}{2}} \sqrt {b x +a}}{3 b}-\frac {5 \left (a d -b c \right ) \left (\frac {\left (d x +c \right )^{\frac {3}{2}} \sqrt {b x +a}}{2 b}-\frac {3 \left (a d -b c \right ) \left (\frac {\sqrt {b x +a}\, \sqrt {d x +c}}{b}-\frac {\left (a d -b c \right ) \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d x}{\sqrt {b d}}+\sqrt {b d \,x^{2}+\left (a d +b c \right ) x +a c}\right )}{2 b \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}}\right )}{4 b}\right )}{6 b}\right )}{8 d}\right )}{10 d}\) | \(239\) |
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 = 1.02, size = 702, normalized size = 3.13 \begin {gather*} \left [-\frac {15 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\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 (128 \, b^{5} d^{5} x^{4} - 15 \, b^{5} c^{4} d + 70 \, a b^{4} c^{3} d^{2} + 128 \, a^{2} b^{3} c^{2} d^{3} - 70 \, a^{3} b^{2} c d^{4} + 15 \, a^{4} b d^{5} + 16 \, {\left (21 \, b^{5} c d^{4} + 11 \, a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (31 \, b^{5} c^{2} d^{3} + 64 \, a b^{4} c d^{4} + a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (5 \, b^{5} c^{3} d^{2} + 233 \, a b^{4} c^{2} d^{3} + 23 \, a^{2} b^{3} c d^{4} - 5 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2560 \, b^{4} d^{3}}, -\frac {15 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\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 (128 \, b^{5} d^{5} x^{4} - 15 \, b^{5} c^{4} d + 70 \, a b^{4} c^{3} d^{2} + 128 \, a^{2} b^{3} c^{2} d^{3} - 70 \, a^{3} b^{2} c d^{4} + 15 \, a^{4} b d^{5} + 16 \, {\left (21 \, b^{5} c d^{4} + 11 \, a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (31 \, b^{5} c^{2} d^{3} + 64 \, a b^{4} c d^{4} + a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (5 \, b^{5} c^{3} d^{2} + 233 \, a b^{4} c^{2} d^{3} + 23 \, a^{2} b^{3} c d^{4} - 5 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{1280 \, b^{4} 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 {5}{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. 1962 vs.
\(2 (180) = 360\).
time = 1.75, size = 1962, normalized size = 8.76 \begin {gather*} \text {Too large to display} \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 {\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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