Optimal. Leaf size=278 \[ -\frac {7 b \sqrt {c+d x} (15 b c-7 a d) (b c-a d)}{24 a^4 (a+b x)^{3/2}}-\frac {\sqrt {c+d x} (21 b c-11 a d) (b c-a d)}{8 a^3 x (a+b x)^{3/2}}+\frac {3 c \sqrt {c+d x} (b c-a d)}{4 a^2 x^2 (a+b x)^{3/2}}+\frac {5 (b c-a d) \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{11/2} \sqrt {c}}-\frac {b \sqrt {c+d x} \left (113 a^2 d^2-420 a b c d+315 b^2 c^2\right )}{24 a^5 \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}} \]
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Rubi [A] time = 0.37, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {98, 149, 151, 152, 12, 93, 208} \begin {gather*} -\frac {b \sqrt {c+d x} \left (113 a^2 d^2-420 a b c d+315 b^2 c^2\right )}{24 a^5 \sqrt {a+b x}}+\frac {5 (b c-a d) \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{11/2} \sqrt {c}}+\frac {3 c \sqrt {c+d x} (b c-a d)}{4 a^2 x^2 (a+b x)^{3/2}}-\frac {7 b \sqrt {c+d x} (15 b c-7 a d) (b c-a d)}{24 a^4 (a+b x)^{3/2}}-\frac {\sqrt {c+d x} (21 b c-11 a d) (b c-a d)}{8 a^3 x (a+b x)^{3/2}}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 98
Rule 149
Rule 151
Rule 152
Rule 208
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^{5/2}} \, dx &=-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}-\frac {\int \frac {\sqrt {c+d x} \left (\frac {9}{2} c (b c-a d)+3 d (b c-a d) x\right )}{x^3 (a+b x)^{5/2}} \, dx}{3 a}\\ &=\frac {3 c (b c-a d) \sqrt {c+d x}}{4 a^2 x^2 (a+b x)^{3/2}}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}-\frac {\int \frac {-\frac {3}{4} c (21 b c-11 a d) (b c-a d)-\frac {3}{2} d (9 b c-4 a d) (b c-a d) x}{x^2 (a+b x)^{5/2} \sqrt {c+d x}} \, dx}{6 a^2}\\ &=\frac {3 c (b c-a d) \sqrt {c+d x}}{4 a^2 x^2 (a+b x)^{3/2}}-\frac {(21 b c-11 a d) (b c-a d) \sqrt {c+d x}}{8 a^3 x (a+b x)^{3/2}}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}+\frac {\int \frac {-\frac {15}{8} c (b c-a d) \left (21 b^2 c^2-14 a b c d+a^2 d^2\right )-\frac {3}{2} b c d (21 b c-11 a d) (b c-a d) x}{x (a+b x)^{5/2} \sqrt {c+d x}} \, dx}{6 a^3 c}\\ &=-\frac {7 b (15 b c-7 a d) (b c-a d) \sqrt {c+d x}}{24 a^4 (a+b x)^{3/2}}+\frac {3 c (b c-a d) \sqrt {c+d x}}{4 a^2 x^2 (a+b x)^{3/2}}-\frac {(21 b c-11 a d) (b c-a d) \sqrt {c+d x}}{8 a^3 x (a+b x)^{3/2}}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}+\frac {\int \frac {-\frac {45}{16} c (b c-a d)^2 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right )-\frac {21}{8} b c d (15 b c-7 a d) (b c-a d)^2 x}{x (a+b x)^{3/2} \sqrt {c+d x}} \, dx}{9 a^4 c (b c-a d)}\\ &=-\frac {7 b (15 b c-7 a d) (b c-a d) \sqrt {c+d x}}{24 a^4 (a+b x)^{3/2}}+\frac {3 c (b c-a d) \sqrt {c+d x}}{4 a^2 x^2 (a+b x)^{3/2}}-\frac {(21 b c-11 a d) (b c-a d) \sqrt {c+d x}}{8 a^3 x (a+b x)^{3/2}}-\frac {b \left (315 b^2 c^2-420 a b c d+113 a^2 d^2\right ) \sqrt {c+d x}}{24 a^5 \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}+\frac {2 \int -\frac {45 c (b c-a d)^3 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right )}{32 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{9 a^5 c (b c-a d)^2}\\ &=-\frac {7 b (15 b c-7 a d) (b c-a d) \sqrt {c+d x}}{24 a^4 (a+b x)^{3/2}}+\frac {3 c (b c-a d) \sqrt {c+d x}}{4 a^2 x^2 (a+b x)^{3/2}}-\frac {(21 b c-11 a d) (b c-a d) \sqrt {c+d x}}{8 a^3 x (a+b x)^{3/2}}-\frac {b \left (315 b^2 c^2-420 a b c d+113 a^2 d^2\right ) \sqrt {c+d x}}{24 a^5 \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}-\frac {\left (5 (b c-a d) \left (21 b^2 c^2-14 a b c d+a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 a^5}\\ &=-\frac {7 b (15 b c-7 a d) (b c-a d) \sqrt {c+d x}}{24 a^4 (a+b x)^{3/2}}+\frac {3 c (b c-a d) \sqrt {c+d x}}{4 a^2 x^2 (a+b x)^{3/2}}-\frac {(21 b c-11 a d) (b c-a d) \sqrt {c+d x}}{8 a^3 x (a+b x)^{3/2}}-\frac {b \left (315 b^2 c^2-420 a b c d+113 a^2 d^2\right ) \sqrt {c+d x}}{24 a^5 \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}-\frac {\left (5 (b c-a d) \left (21 b^2 c^2-14 a b c d+a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{8 a^5}\\ &=-\frac {7 b (15 b c-7 a d) (b c-a d) \sqrt {c+d x}}{24 a^4 (a+b x)^{3/2}}+\frac {3 c (b c-a d) \sqrt {c+d x}}{4 a^2 x^2 (a+b x)^{3/2}}-\frac {(21 b c-11 a d) (b c-a d) \sqrt {c+d x}}{8 a^3 x (a+b x)^{3/2}}-\frac {b \left (315 b^2 c^2-420 a b c d+113 a^2 d^2\right ) \sqrt {c+d x}}{24 a^5 \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}+\frac {5 (b c-a d) \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{11/2} \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 199, normalized size = 0.72 \begin {gather*} \frac {2 a^{7/2} x (c+d x)^{7/2} (9 b c-a d)-8 a^{9/2} c (c+d x)^{7/2}-x^2 \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \left (3 a^{5/2} (c+d x)^{5/2}+5 x (b c-a d) \left (\sqrt {a} \sqrt {c+d x} (4 a c+a d x+3 b c x)-3 c^{3/2} (a+b x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )\right )}{24 a^{11/2} c^2 x^3 (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.04, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 12.23, size = 848, normalized size = 3.05 \begin {gather*} \left [-\frac {15 \, {\left ({\left (21 \, b^{5} c^{3} - 35 \, a b^{4} c^{2} d + 15 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{5} + 2 \, {\left (21 \, a b^{4} c^{3} - 35 \, a^{2} b^{3} c^{2} d + 15 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} + {\left (21 \, a^{2} b^{3} c^{3} - 35 \, a^{3} b^{2} c^{2} d + 15 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3}\right )} \sqrt {a c} \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 (8 \, a^{5} c^{3} + {\left (315 \, a b^{4} c^{3} - 420 \, a^{2} b^{3} c^{2} d + 113 \, a^{3} b^{2} c d^{2}\right )} x^{4} + 2 \, {\left (210 \, a^{2} b^{3} c^{3} - 287 \, a^{3} b^{2} c^{2} d + 81 \, a^{4} b c d^{2}\right )} x^{3} + 3 \, {\left (21 \, a^{3} b^{2} c^{3} - 32 \, a^{4} b c^{2} d + 11 \, a^{5} c d^{2}\right )} x^{2} - 2 \, {\left (9 \, a^{4} b c^{3} - 13 \, a^{5} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (a^{6} b^{2} c x^{5} + 2 \, a^{7} b c x^{4} + a^{8} c x^{3}\right )}}, -\frac {15 \, {\left ({\left (21 \, b^{5} c^{3} - 35 \, a b^{4} c^{2} d + 15 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{5} + 2 \, {\left (21 \, a b^{4} c^{3} - 35 \, a^{2} b^{3} c^{2} d + 15 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} + {\left (21 \, a^{2} b^{3} c^{3} - 35 \, a^{3} b^{2} c^{2} d + 15 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3}\right )} \sqrt {-a c} \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 (8 \, a^{5} c^{3} + {\left (315 \, a b^{4} c^{3} - 420 \, a^{2} b^{3} c^{2} d + 113 \, a^{3} b^{2} c d^{2}\right )} x^{4} + 2 \, {\left (210 \, a^{2} b^{3} c^{3} - 287 \, a^{3} b^{2} c^{2} d + 81 \, a^{4} b c d^{2}\right )} x^{3} + 3 \, {\left (21 \, a^{3} b^{2} c^{3} - 32 \, a^{4} b c^{2} d + 11 \, a^{5} c d^{2}\right )} x^{2} - 2 \, {\left (9 \, a^{4} b c^{3} - 13 \, a^{5} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (a^{6} b^{2} c x^{5} + 2 \, a^{7} b c x^{4} + a^{8} c x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 1009, normalized size = 3.63 \begin {gather*} -\frac {\sqrt {d x +c}\, \left (15 a^{3} b^{2} d^{3} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-225 a^{2} b^{3} c \,d^{2} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+525 a \,b^{4} c^{2} d \,x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-315 b^{5} c^{3} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+30 a^{4} b \,d^{3} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-450 a^{3} b^{2} c \,d^{2} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+1050 a^{2} b^{3} c^{2} d \,x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-630 a \,b^{4} c^{3} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+15 a^{5} d^{3} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-225 a^{4} b c \,d^{2} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+525 a^{3} b^{2} c^{2} d \,x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-315 a^{2} b^{3} c^{3} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+226 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b^{2} d^{2} x^{4}-840 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{3} c d \,x^{4}+630 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{4} c^{2} x^{4}+324 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} b \,d^{2} x^{3}-1148 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b^{2} c d \,x^{3}+840 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{3} c^{2} x^{3}+66 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{4} d^{2} x^{2}-192 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} b c d \,x^{2}+126 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b^{2} c^{2} x^{2}+52 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{4} c d x -36 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} b \,c^{2} x +16 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{4} c^{2}\right )}{48 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, \left (b x +a \right )^{\frac {3}{2}} a^{5} x^{3}} \end {gather*}
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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{5/2}}{x^4\,{\left (a+b\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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