Optimal. Leaf size=454 \[ -\frac {2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b d \left (8 A c^2 d+b^2 B e-b c (4 B d+7 A e)\right )+\left (16 A c^3 d^2+2 b^3 B e^2+b^2 c e (3 B d+A e)-8 b c^2 d (B d+2 A e)\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {2 \left (16 A c^3 d^2+2 b^3 B e^2+b^2 c e (3 B d+A e)-8 b c^2 d (B d+2 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{3/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 d (c d-b e) \left (16 A c^2 d-b^2 B e-8 b c (B d+A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{3/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \]
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Rubi [A]
time = 0.37, antiderivative size = 454, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {832, 834, 857,
729, 113, 111, 118, 117} \begin {gather*} \frac {2 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (-8 b c (A e+B d)+16 A c^2 d+b^2 (-B) e\right ) F\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{3/2} \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 c e (A e+3 B d)-8 b c^2 d (2 A e+B d)+16 A c^3 d^2+2 b^3 B e^2\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{3/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 (d+e x)^{3/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b d \left (-b c (7 A e+4 B d)+8 A c^2 d+b^2 B e\right )+x \left (b^2 c e (A e+3 B d)-8 b c^2 d (2 A e+B d)+16 A c^3 d^2+2 b^3 B e^2\right )\right )}{3 b^4 c \sqrt {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 111
Rule 113
Rule 117
Rule 118
Rule 729
Rule 832
Rule 834
Rule 857
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \int \frac {\sqrt {d+e x} \left (\frac {1}{2} d \left (4 b B c d-8 A c^2 d-b^2 B e+7 A b c e\right )-\frac {1}{2} e \left (2 A c^2 d-2 b^2 B e-b c (B d+A e)\right ) x\right )}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2 c}\\ &=-\frac {2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b d \left (8 A c^2 d+b^2 B e-b c (4 B d+7 A e)\right )+\left (16 A c^3 d^2+2 b^3 B e^2+b^2 c e (3 B d+A e)-8 b c^2 d (B d+2 A e)\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {4 \int \frac {-\frac {1}{4} b d e \left (4 b B c d-8 A c^2 d-b^2 B e+7 A b c e\right )+\frac {1}{4} e \left (16 A c^3 d^2+2 b^3 B e^2+b^2 c e (3 B d+A e)-8 b c^2 d (B d+2 A e)\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 c}\\ &=-\frac {2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b d \left (8 A c^2 d+b^2 B e-b c (4 B d+7 A e)\right )+\left (16 A c^3 d^2+2 b^3 B e^2+b^2 c e (3 B d+A e)-8 b c^2 d (B d+2 A e)\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}+\frac {\left (d (c d-b e) \left (16 A c^2 d-b^2 B e-8 b c (B d+A e)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 c}-\frac {\left (16 A c^3 d^2+2 b^3 B e^2+b^2 c e (3 B d+A e)-8 b c^2 d (B d+2 A e)\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 b^4 c}\\ &=-\frac {2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b d \left (8 A c^2 d+b^2 B e-b c (4 B d+7 A e)\right )+\left (16 A c^3 d^2+2 b^3 B e^2+b^2 c e (3 B d+A e)-8 b c^2 d (B d+2 A e)\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}+\frac {\left (d (c d-b e) \left (16 A c^2 d-b^2 B e-8 b c (B d+A e)\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{3 b^4 c \sqrt {b x+c x^2}}-\frac {\left (\left (16 A c^3 d^2+2 b^3 B e^2+b^2 c e (3 B d+A e)-8 b c^2 d (B d+2 A e)\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{3 b^4 c \sqrt {b x+c x^2}}\\ &=-\frac {2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b d \left (8 A c^2 d+b^2 B e-b c (4 B d+7 A e)\right )+\left (16 A c^3 d^2+2 b^3 B e^2+b^2 c e (3 B d+A e)-8 b c^2 d (B d+2 A e)\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {\left (\left (16 A c^3 d^2+2 b^3 B e^2+b^2 c e (3 B d+A e)-8 b c^2 d (B d+2 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x}\right ) \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {x} \sqrt {1+\frac {c x}{b}}} \, dx}{3 b^4 c \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (d (c d-b e) \left (16 A c^2 d-b^2 B e-8 b c (B d+A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}} \, dx}{3 b^4 c \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=-\frac {2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b d \left (8 A c^2 d+b^2 B e-b c (4 B d+7 A e)\right )+\left (16 A c^3 d^2+2 b^3 B e^2+b^2 c e (3 B d+A e)-8 b c^2 d (B d+2 A e)\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {2 \left (16 A c^3 d^2+2 b^3 B e^2+b^2 c e (3 B d+A e)-8 b c^2 d (B d+2 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{3/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 d (c d-b e) \left (16 A c^2 d-b^2 B e-8 b c (B d+A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{3/2} \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 22.45, size = 452, normalized size = 1.00 \begin {gather*} -\frac {2 \left (b (d+e x) \left (b (b B-A c) (c d-b e)^2 x^2+(c d-b e) \left (-8 A c^2 d+2 b^2 B e+b c (5 B d+A e)\right ) x^2 (b+c x)+A b c d^2 (b+c x)^2+c d (3 b B d-8 A c d+7 A b e) x (b+c x)^2\right )+\sqrt {\frac {b}{c}} x (b+c x) \left (\sqrt {\frac {b}{c}} \left (16 A c^3 d^2+2 b^3 B e^2+b^2 c e (3 B d+A e)-8 b c^2 d (B d+2 A e)\right ) (b+c x) (d+e x)+i b e \left (16 A c^3 d^2+2 b^3 B e^2+b^2 c e (3 B d+A e)-8 b c^2 d (B d+2 A e)\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e (c d-b e) \left (8 A c^2 d-2 b^2 B e-b c (4 B d+A e)\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )\right )\right )}{3 b^5 c (x (b+c x))^{3/2} \sqrt {d+e x}} \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. \(2643\) vs.
\(2(400)=800\).
time = 0.73, size = 2644, normalized size = 5.82
method | result | size |
elliptic | \(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (\frac {2 \left (A \,b^{2} c \,e^{2}-2 A b \,c^{2} d e +A \,c^{3} d^{2}-b^{3} B \,e^{2}+2 B \,b^{2} c d e -B b \,c^{2} d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{3 b^{3} c^{3} \left (\frac {b}{c}+x \right )^{2}}+\frac {2 \left (c e \,x^{2}+c d x \right ) \left (A \,b^{2} c \,e^{2}-9 A b \,c^{2} d e +8 A \,c^{3} d^{2}+2 b^{3} B \,e^{2}+3 B \,b^{2} c d e -5 B b \,c^{2} d^{2}\right )}{3 b^{4} c^{2} \sqrt {\left (\frac {b}{c}+x \right ) \left (c e \,x^{2}+c d x \right )}}-\frac {2 A \,d^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{3 b^{3} x^{2}}-\frac {2 \left (c e \,x^{2}+b e x +c d x +b d \right ) d \left (7 A b e -8 A c d +3 B b d \right )}{3 b^{4} \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}+\frac {2 \left (\frac {B \,e^{3}}{c^{2}}+\frac {\left (A \,b^{2} c \,e^{2}-2 A b \,c^{2} d e +A \,c^{3} d^{2}-b^{3} B \,e^{2}+2 B \,b^{2} c d e -B b \,c^{2} d^{2}\right ) e}{3 c^{2} b^{3}}-\frac {\left (A \,b^{2} c \,e^{2}-9 A b \,c^{2} d e +8 A \,c^{3} d^{2}+2 b^{3} B \,e^{2}+3 B \,b^{2} c d e -5 B b \,c^{2} d^{2}\right ) \left (b e -c d \right )}{3 c^{2} b^{4}}-\frac {d \left (A \,b^{2} c \,e^{2}-9 A b \,c^{2} d e +8 A \,c^{3} d^{2}+2 b^{3} B \,e^{2}+3 B \,b^{2} c d e -5 B b \,c^{2} d^{2}\right )}{3 c \,b^{4}}-\frac {d^{2} A c e}{3 b^{3}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}+\frac {2 \left (-\frac {\left (A \,b^{2} c \,e^{2}-9 A b \,c^{2} d e +8 A \,c^{3} d^{2}+2 b^{3} B \,e^{2}+3 B \,b^{2} c d e -5 B b \,c^{2} d^{2}\right ) e}{3 c \,b^{4}}+\frac {c d e \left (7 A b e -8 A c d +3 B b d \right )}{3 b^{4}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) \EllipticE \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d \EllipticF \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\) | \(923\) |
default | \(\text {Expression too large to display}\) | \(2644\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.39, size = 1037, normalized size = 2.28 \begin {gather*} -\frac {2 \, {\left ({\left (8 \, {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{3} x^{4} + 16 \, {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{3} x^{3} + 8 \, {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{3} x^{2} - {\left ({\left (2 \, B b^{4} c^{2} + A b^{3} c^{3}\right )} x^{4} + 2 \, {\left (2 \, B b^{5} c + A b^{4} c^{2}\right )} x^{3} + {\left (2 \, B b^{6} + A b^{5} c\right )} x^{2}\right )} e^{3} - 2 \, {\left ({\left (B b^{3} c^{3} + 3 \, A b^{2} c^{4}\right )} d x^{4} + 2 \, {\left (B b^{4} c^{2} + 3 \, A b^{3} c^{3}\right )} d x^{3} + {\left (B b^{5} c + 3 \, A b^{4} c^{2}\right )} d x^{2}\right )} e^{2} - {\left ({\left (7 \, B b^{2} c^{4} - 24 \, A b c^{5}\right )} d^{2} x^{4} + 2 \, {\left (7 \, B b^{3} c^{3} - 24 \, A b^{2} c^{4}\right )} d^{2} x^{3} + {\left (7 \, B b^{4} c^{2} - 24 \, A b^{3} c^{3}\right )} d^{2} x^{2}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) - 3 \, {\left ({\left ({\left (2 \, B b^{3} c^{3} + A b^{2} c^{4}\right )} x^{4} + 2 \, {\left (2 \, B b^{4} c^{2} + A b^{3} c^{3}\right )} x^{3} + {\left (2 \, B b^{5} c + A b^{4} c^{2}\right )} x^{2}\right )} e^{3} + {\left ({\left (3 \, B b^{2} c^{4} - 16 \, A b c^{5}\right )} d x^{4} + 2 \, {\left (3 \, B b^{3} c^{3} - 16 \, A b^{2} c^{4}\right )} d x^{3} + {\left (3 \, B b^{4} c^{2} - 16 \, A b^{3} c^{3}\right )} d x^{2}\right )} e^{2} - 8 \, {\left ({\left (B b c^{5} - 2 \, A c^{6}\right )} d^{2} x^{4} + 2 \, {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{2} x^{3} + {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{2} x^{2}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right ) - 3 \, \sqrt {c x^{2} + b x} {\left ({\left ({\left (2 \, B b^{3} c^{3} + A b^{2} c^{4}\right )} x^{3} + {\left (B b^{4} c^{2} + 2 \, A b^{3} c^{3}\right )} x^{2}\right )} e^{3} - {\left (7 \, A b^{3} c^{3} d x - {\left (3 \, B b^{2} c^{4} - 16 \, A b c^{5}\right )} d x^{3} - 5 \, {\left (B b^{3} c^{3} - 5 \, A b^{2} c^{4}\right )} d x^{2}\right )} e^{2} - {\left (A b^{3} c^{3} d^{2} + 8 \, {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{2} x^{3} + 12 \, {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{2} x^{2} + 3 \, {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{2} x\right )} e\right )} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{9 \, {\left (b^{4} c^{5} x^{4} + 2 \, b^{5} c^{4} x^{3} + b^{6} c^{3} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 (A+B\,x\right )\,{\left (d+e\,x\right )}^{5/2}}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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