Optimal. Leaf size=391 \[ -2 a^{5/2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-3 a^2 d^2-16 a b c d+3 b^2 c^2\right )}{48 d^2}+\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (3 a^3 d^3+109 a^2 b c d^2-19 a b^2 c^2 d+3 b^3 c^3\right )}{192 b d^2}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-3 a^4 d^4+22 a^3 b c d^3+128 a^2 b^2 c^2 d^2-22 a b^3 c^3 d+3 b^4 c^4\right )}{128 b^2 d^2}+\frac {(a d+b c) \left (3 a^4 d^4-28 a^3 b c d^3+178 a^2 b^2 c^2 d^2-28 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{5/2} d^{5/2}}+\frac {1}{5} (a+b x)^{5/2} (c+d x)^{5/2}+\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{8 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.43, antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {101, 154, 157, 63, 217, 206, 93, 208} \[ -\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-3 a^2 d^2-16 a b c d+3 b^2 c^2\right )}{48 d^2}+\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (109 a^2 b c d^2+3 a^3 d^3-19 a b^2 c^2 d+3 b^3 c^3\right )}{192 b d^2}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4-22 a b^3 c^3 d+3 b^4 c^4\right )}{128 b^2 d^2}+\frac {(a d+b c) \left (178 a^2 b^2 c^2 d^2-28 a^3 b c d^3+3 a^4 d^4-28 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{5/2} d^{5/2}}-2 a^{5/2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {1}{5} (a+b x)^{5/2} (c+d x)^{5/2}+\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{8 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 93
Rule 101
Rule 154
Rule 157
Rule 206
Rule 208
Rule 217
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x} \, dx &=\frac {1}{5} (a+b x)^{5/2} (c+d x)^{5/2}-\frac {1}{5} \int \frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (-5 a c-\frac {5}{2} (b c+a d) x\right )}{x} \, dx\\ &=\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 d}+\frac {1}{5} (a+b x)^{5/2} (c+d x)^{5/2}-\frac {\int \frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-20 a^2 c d+\frac {5}{4} \left (3 b^2 c^2-16 a b c d-3 a^2 d^2\right ) x\right )}{x} \, dx}{20 d}\\ &=-\frac {\left (3 b^2 c^2-16 a b c d-3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{48 d^2}+\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 d}+\frac {1}{5} (a+b x)^{5/2} (c+d x)^{5/2}-\frac {\int \frac {(c+d x)^{3/2} \left (-60 a^3 c d^2-\frac {5}{8} \left (3 b^3 c^3-19 a b^2 c^2 d+109 a^2 b c d^2+3 a^3 d^3\right ) x\right )}{x \sqrt {a+b x}} \, dx}{60 d^2}\\ &=\frac {\left (3 b^3 c^3-19 a b^2 c^2 d+109 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 b d^2}-\frac {\left (3 b^2 c^2-16 a b c d-3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{48 d^2}+\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 d}+\frac {1}{5} (a+b x)^{5/2} (c+d x)^{5/2}-\frac {\int \frac {\sqrt {c+d x} \left (-120 a^3 b c^2 d^2-\frac {15}{16} \left (3 b^4 c^4-22 a b^3 c^3 d+128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4\right ) x\right )}{x \sqrt {a+b x}} \, dx}{120 b d^2}\\ &=\frac {\left (3 b^4 c^4-22 a b^3 c^3 d+128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^2 d^2}+\frac {\left (3 b^3 c^3-19 a b^2 c^2 d+109 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 b d^2}-\frac {\left (3 b^2 c^2-16 a b c d-3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{48 d^2}+\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 d}+\frac {1}{5} (a+b x)^{5/2} (c+d x)^{5/2}-\frac {\int \frac {-120 a^3 b^2 c^3 d^2-\frac {15}{32} (b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2-28 a^3 b c d^3+3 a^4 d^4\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{120 b^2 d^2}\\ &=\frac {\left (3 b^4 c^4-22 a b^3 c^3 d+128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^2 d^2}+\frac {\left (3 b^3 c^3-19 a b^2 c^2 d+109 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 b d^2}-\frac {\left (3 b^2 c^2-16 a b c d-3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{48 d^2}+\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 d}+\frac {1}{5} (a+b x)^{5/2} (c+d x)^{5/2}+\left (a^3 c^3\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {\left ((b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2-28 a^3 b c d^3+3 a^4 d^4\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 b^2 d^2}\\ &=\frac {\left (3 b^4 c^4-22 a b^3 c^3 d+128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^2 d^2}+\frac {\left (3 b^3 c^3-19 a b^2 c^2 d+109 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 b d^2}-\frac {\left (3 b^2 c^2-16 a b c d-3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{48 d^2}+\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 d}+\frac {1}{5} (a+b x)^{5/2} (c+d x)^{5/2}+\left (2 a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )+\frac {\left ((b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2-28 a^3 b c d^3+3 a^4 d^4\right )\right ) \operatorname {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^3 d^2}\\ &=\frac {\left (3 b^4 c^4-22 a b^3 c^3 d+128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^2 d^2}+\frac {\left (3 b^3 c^3-19 a b^2 c^2 d+109 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 b d^2}-\frac {\left (3 b^2 c^2-16 a b c d-3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{48 d^2}+\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 d}+\frac {1}{5} (a+b x)^{5/2} (c+d x)^{5/2}-2 a^{5/2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {\left ((b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2-28 a^3 b c d^3+3 a^4 d^4\right )\right ) \operatorname {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^3 d^2}\\ &=\frac {\left (3 b^4 c^4-22 a b^3 c^3 d+128 a^2 b^2 c^2 d^2+22 a^3 b c d^3-3 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^2 d^2}+\frac {\left (3 b^3 c^3-19 a b^2 c^2 d+109 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 b d^2}-\frac {\left (3 b^2 c^2-16 a b c d-3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{48 d^2}+\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 d}+\frac {1}{5} (a+b x)^{5/2} (c+d x)^{5/2}-2 a^{5/2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {(b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2-28 a^3 b c d^3+3 a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{5/2} d^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 3.62, size = 1249, normalized size = 3.19 \[ \frac {\sqrt {c+d x} \left (45 b^5 (c+d x)^2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) c^5-45 b^2 \sqrt {d} (b c-a d)^{5/2} \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c^4-375 a b^4 d (c+d x)^2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) c^4+360 a b d^{3/2} (b c-a d)^{5/2} \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c^3+30 b^2 d^{3/2} (b c-a d)^{5/2} x \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c^3+2250 a^2 b^3 d^2 (c+d x)^2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) c^3-3840 a^{5/2} b d^{5/2} (b c-a d)^{3/2} \sqrt {c+d x} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right ) c^{5/2}+3754 a^2 d^{5/2} (b c-a d)^{5/2} \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c^2+744 b^2 d^{5/2} (b c-a d)^{5/2} x^2 \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c^2+2578 a b d^{5/2} (b c-a d)^{5/2} x \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c^2+2250 a^3 b^2 d^3 (c+d x)^2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) c^2+1008 b^2 d^{7/2} (b c-a d)^{5/2} x^3 \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c+\frac {360 a^3 d^{7/2} (b c-a d)^{5/2} \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c}{b}+2896 a b d^{7/2} (b c-a d)^{5/2} x^2 \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c+2578 a^2 d^{7/2} (b c-a d)^{5/2} x \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c-375 a^4 b d^4 (c+d x)^2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) c+384 b^2 d^{9/2} (b c-a d)^{5/2} x^4 \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2}+1008 a b d^{9/2} (b c-a d)^{5/2} x^3 \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2}+744 a^2 d^{9/2} (b c-a d)^{5/2} x^2 \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2}+\frac {30 a^3 d^{9/2} (b c-a d)^{5/2} x \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2}}{b}+45 a^5 d^5 (c+d x)^2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )-45 a^4 d^{9/2} \sqrt {b c-a d} \sqrt {a+b x} (c+d x)^2 \sqrt {\frac {b (c+d x)}{b c-a d}}\right )}{1920 d^{5/2} (b c-a d)^{5/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 124.57, size = 1801, normalized size = 4.61 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 1116, normalized size = 2.85 \[ \frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (768 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, \sqrt {b d}\, b^{4} d^{4} x^{4}+45 \sqrt {a c}\, a^{5} d^{5} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-375 \sqrt {a c}\, a^{4} b c \,d^{4} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-3840 \sqrt {b d}\, a^{3} b^{2} c^{3} d^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+2250 \sqrt {a c}\, a^{3} b^{2} c^{2} d^{3} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+2250 \sqrt {a c}\, a^{2} b^{3} c^{3} d^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-375 \sqrt {a c}\, a \,b^{4} c^{4} d \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+2016 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, \sqrt {b d}\, a \,b^{3} d^{4} x^{3}+45 \sqrt {a c}\, b^{5} c^{5} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+2016 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, \sqrt {b d}\, b^{4} c \,d^{3} x^{3}+1488 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, \sqrt {b d}\, a^{2} b^{2} d^{4} x^{2}+5792 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, \sqrt {b d}\, a \,b^{3} c \,d^{3} x^{2}+1488 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, \sqrt {b d}\, b^{4} c^{2} d^{2} x^{2}+60 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, \sqrt {b d}\, a^{3} b \,d^{4} x +5156 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, \sqrt {b d}\, a^{2} b^{2} c \,d^{3} x +5156 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, \sqrt {b d}\, a \,b^{3} c^{2} d^{2} x +60 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, \sqrt {b d}\, b^{4} c^{3} d x -90 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, \sqrt {b d}\, a^{4} d^{4}+720 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, \sqrt {b d}\, a^{3} b c \,d^{3}+7508 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, \sqrt {b d}\, a^{2} b^{2} c^{2} d^{2}+720 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, \sqrt {b d}\, a \,b^{3} c^{3} d -90 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {a c}\, \sqrt {b d}\, b^{4} c^{4}\right )}{3840 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________