Optimal. Leaf size=340 \[ -\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a x^4}+\frac {\left (\frac {7 b^2 c}{a}-12 b d-\frac {3 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a x^3}-\frac {\left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^2 x^2}+\frac {\left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^4 c^3 x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{5 x^5}-\frac {(b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{7/2}} \]
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Rubi [A]
time = 0.23, antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {99, 154, 156,
12, 95, 214} \begin {gather*} -\frac {\left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{7/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-15 a^3 d^3+9 a^2 b c d^2-61 a b^2 c^2 d+35 b^3 c^3\right )}{960 a^3 c^2 x^2}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-45 a^4 d^4+30 a^3 b c d^3+36 a^2 b^2 c^2 d^2-190 a b^3 c^3 d+105 b^4 c^4\right )}{1920 a^4 c^3 x}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {7 b^2 c}{a}-\frac {3 a d^2}{c}-12 b d\right )}{240 a x^3}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{5 x^5}-\frac {\sqrt {a+b x} \sqrt {c+d x} (3 a d+b c)}{40 a x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 99
Rule 154
Rule 156
Rule 214
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^6} \, dx &=-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{5 x^5}+\frac {1}{5} \int \frac {\sqrt {c+d x} \left (\frac {1}{2} (b c+3 a d)+2 b d x\right )}{x^5 \sqrt {a+b x}} \, dx\\ &=-\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a x^4}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{5 x^5}+\frac {\int \frac {\frac {1}{4} \left (-7 b^2 c^2+12 a b c d+3 a^2 d^2\right )-\frac {1}{2} b d (3 b c-7 a d) x}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{20 a}\\ &=-\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a x^4}+\frac {\left (\frac {7 b^2 c}{a}-12 b d-\frac {3 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a x^3}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{5 x^5}-\frac {\int \frac {\frac {1}{8} \left (-35 b^3 c^3+61 a b^2 c^2 d-9 a^2 b c d^2+15 a^3 d^3\right )-\frac {1}{2} b d \left (7 b^2 c^2-12 a b c d-3 a^2 d^2\right ) x}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{60 a^2 c}\\ &=-\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a x^4}+\frac {\left (\frac {7 b^2 c}{a}-12 b d-\frac {3 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a x^3}-\frac {\left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^2 x^2}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{5 x^5}+\frac {\int \frac {\frac {1}{16} \left (-105 b^4 c^4+190 a b^3 c^3 d-36 a^2 b^2 c^2 d^2-30 a^3 b c d^3+45 a^4 d^4\right )-\frac {1}{8} b d \left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{120 a^3 c^2}\\ &=-\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a x^4}+\frac {\left (\frac {7 b^2 c}{a}-12 b d-\frac {3 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a x^3}-\frac {\left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^2 x^2}+\frac {\left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^4 c^3 x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{5 x^5}-\frac {\int -\frac {15 (b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )}{32 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{120 a^4 c^3}\\ &=-\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a x^4}+\frac {\left (\frac {7 b^2 c}{a}-12 b d-\frac {3 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a x^3}-\frac {\left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^2 x^2}+\frac {\left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^4 c^3 x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{5 x^5}+\frac {\left ((b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 a^4 c^3}\\ &=-\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a x^4}+\frac {\left (\frac {7 b^2 c}{a}-12 b d-\frac {3 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a x^3}-\frac {\left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^2 x^2}+\frac {\left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^4 c^3 x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{5 x^5}+\frac {\left ((b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{128 a^4 c^3}\\ &=-\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a x^4}+\frac {\left (\frac {7 b^2 c}{a}-12 b d-\frac {3 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a x^3}-\frac {\left (35 b^3 c^3-61 a b^2 c^2 d+9 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^2 x^2}+\frac {\left (105 b^4 c^4-190 a b^3 c^3 d+36 a^2 b^2 c^2 d^2+30 a^3 b c d^3-45 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^4 c^3 x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{5 x^5}-\frac {(b c-a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.57, size = 258, normalized size = 0.76 \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 b^4 c^4 x^4+10 a b^3 c^3 x^3 (7 c+19 d x)-2 a^2 b^2 c^2 x^2 \left (28 c^2+61 c d x+18 d^2 x^2\right )+6 a^3 b c x \left (8 c^3+16 c^2 d x+3 c d^2 x^2-5 d^3 x^3\right )+3 a^4 \left (128 c^4+176 c^3 d x+8 c^2 d^2 x^2-10 c d^3 x^3+15 d^4 x^4\right )\right )}{1920 a^4 c^3 x^5}+\frac {(-b c+a d)^3 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{7/2}} \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. \(812\) vs.
\(2(296)=592\).
time = 0.07, size = 813, normalized size = 2.39
method | result | size |
default | \(\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (45 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{5} d^{5} x^{5}-45 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{4} b c \,d^{4} x^{5}-30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} b^{2} c^{2} d^{3} x^{5}-90 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b^{3} c^{3} d^{2} x^{5}+225 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{4} c^{4} d \,x^{5}-105 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{5} c^{5} x^{5}-90 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} d^{4} x^{4}+60 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b c \,d^{3} x^{4}+72 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{2} d^{2} x^{4}-380 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{3} c^{3} d \,x^{4}+210 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{4} c^{4} x^{4}+60 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c \,d^{3} x^{3}-36 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b \,c^{2} d^{2} x^{3}+244 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{3} d \,x^{3}-140 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{3} c^{4} x^{3}-48 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c^{2} d^{2} x^{2}-192 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b \,c^{3} d \,x^{2}+112 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b^{2} c^{4} x^{2}-1056 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c^{3} d x -96 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} b \,c^{4} x -768 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{4} c^{4} \sqrt {a c}\right )}{3840 a^{4} c^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{5} \sqrt {a c}}\) | \(813\) |
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 = 4.53, size = 732, normalized size = 2.15 \begin {gather*} \left [-\frac {15 \, {\left (7 \, b^{5} c^{5} - 15 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + 3 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \sqrt {a c} x^{5} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (384 \, a^{5} c^{5} - {\left (105 \, a b^{4} c^{5} - 190 \, a^{2} b^{3} c^{4} d + 36 \, a^{3} b^{2} c^{3} d^{2} + 30 \, a^{4} b c^{2} d^{3} - 45 \, a^{5} c d^{4}\right )} x^{4} + 2 \, {\left (35 \, a^{2} b^{3} c^{5} - 61 \, a^{3} b^{2} c^{4} d + 9 \, a^{4} b c^{3} d^{2} - 15 \, a^{5} c^{2} d^{3}\right )} x^{3} - 8 \, {\left (7 \, a^{3} b^{2} c^{5} - 12 \, a^{4} b c^{4} d - 3 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (a^{4} b c^{5} + 11 \, a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, a^{5} c^{4} x^{5}}, \frac {15 \, {\left (7 \, b^{5} c^{5} - 15 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + 3 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \sqrt {-a c} x^{5} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (384 \, a^{5} c^{5} - {\left (105 \, a b^{4} c^{5} - 190 \, a^{2} b^{3} c^{4} d + 36 \, a^{3} b^{2} c^{3} d^{2} + 30 \, a^{4} b c^{2} d^{3} - 45 \, a^{5} c d^{4}\right )} x^{4} + 2 \, {\left (35 \, a^{2} b^{3} c^{5} - 61 \, a^{3} b^{2} c^{4} d + 9 \, a^{4} b c^{3} d^{2} - 15 \, a^{5} c^{2} d^{3}\right )} x^{3} - 8 \, {\left (7 \, a^{3} b^{2} c^{5} - 12 \, a^{4} b c^{4} d - 3 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (a^{4} b c^{5} + 11 \, a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, a^{5} c^{4} x^{5}}\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 {\sqrt {a + b x} \left (c + d x\right )^{\frac {3}{2}}}{x^{6}}\, 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. 5870 vs.
\(2 (296) = 592\).
time = 29.12, size = 5870, normalized size = 17.26 \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 \frac {\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{3/2}}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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