Optimal. Leaf size=354 \[ -\frac {2 \left (c d^2 (8 c d-7 b e)+e \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{5 d e^3 (c d-b e) (d+e x)^{3/2}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}+\frac {2 \sqrt {-b} \sqrt {c} \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{5 d e^4 (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {16 \sqrt {-b} \sqrt {c} (2 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{5 e^4 \sqrt {d+e x} \sqrt {b x+c x^2}} \]
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Rubi [A]
time = 0.49, antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {746, 824, 857,
729, 113, 111, 118, 117} \begin {gather*} \frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{5 d e^4 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)}-\frac {16 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (2 c d-b e) F\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{5 e^4 \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {b x+c x^2} \left (e x \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )+c d^2 (8 c d-7 b e)\right )}{5 d e^3 (d+e x)^{3/2} (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 111
Rule 113
Rule 117
Rule 118
Rule 729
Rule 746
Rule 824
Rule 857
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx &=-\frac {2 \left (b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}+\frac {3 \int \frac {(b+2 c x) \sqrt {b x+c x^2}}{(d+e x)^{5/2}} \, dx}{5 e}\\ &=-\frac {2 \left (c d^2 (8 c d-7 b e)+e \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{5 d e^3 (c d-b e) (d+e x)^{3/2}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}-\frac {2 \int \frac {-\frac {1}{2} b c d (8 c d-7 b e)-\frac {1}{2} c \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{5 d e^3 (c d-b e)}\\ &=-\frac {2 \left (c d^2 (8 c d-7 b e)+e \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{5 d e^3 (c d-b e) (d+e x)^{3/2}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}-\frac {(8 c (2 c d-b e)) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{5 e^4}+\frac {\left (c \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{5 d e^4 (c d-b e)}\\ &=-\frac {2 \left (c d^2 (8 c d-7 b e)+e \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{5 d e^3 (c d-b e) (d+e x)^{3/2}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}-\frac {\left (8 c (2 c d-b e) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{5 e^4 \sqrt {b x+c x^2}}+\frac {\left (c \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{5 d e^4 (c d-b e) \sqrt {b x+c x^2}}\\ &=-\frac {2 \left (c d^2 (8 c d-7 b e)+e \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{5 d e^3 (c d-b e) (d+e x)^{3/2}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}+\frac {\left (c \left (16 c^2 d^2-16 b c d e+b^2 e^2\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}{5 d e^4 (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left (8 c (2 c d-b e) \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}{5 e^4 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=-\frac {2 \left (c d^2 (8 c d-7 b e)+e \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{5 d e^3 (c d-b e) (d+e x)^{3/2}}-\frac {2 \left (b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}+\frac {2 \sqrt {-b} \sqrt {c} \left (16 c^2 d^2-16 b c d e+b^2 e^2\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 )}{5 d e^4 (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {16 \sqrt {-b} \sqrt {c} (2 c d-b e) \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 )}{5 e^4 \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 = 10.87, size = 369, normalized size = 1.04 \begin {gather*} -\frac {2 (x (b+c x))^{3/2} \left (b e x (b+c x) \left (b^2 e^4 x^2-b c d e \left (7 d^2+16 d e x+11 e^2 x^2\right )+c^2 d^2 \left (8 d^2+18 d e x+11 e^2 x^2\right )\right )-\sqrt {\frac {b}{c}} c (d+e x)^2 \left (\sqrt {\frac {b}{c}} \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) (b+c x) (d+e x)+i b e \left (16 c^2 d^2-16 b c d e+b^2 e^2\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 \left (8 c^2 d^2-9 b c d e+b^2 e^2\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 )}{5 b d e^4 (c d-b e) x^2 (b+c x)^2 (d+e x)^{5/2}} \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. \(1884\) vs.
\(2(300)=600\).
time = 0.47, size = 1885, normalized size = 5.32
method | result | size |
elliptic | \(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (\frac {2 d \left (b e -c d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{5 e^{6} \left (x +\frac {d}{e}\right )^{3}}-\frac {4 \left (b e -2 c d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{5 e^{5} \left (x +\frac {d}{e}\right )^{2}}+\frac {2 \left (c e \,x^{2}+b e x \right ) \left (b^{2} e^{2}-11 b c d e +11 d^{2} c^{2}\right )}{5 d \left (b e -c d \right ) e^{4} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 \left (\frac {c \left (2 b e -3 c d \right )}{e^{4}}-\frac {2 c \left (b e -2 c d \right )}{5 e^{4}}+\frac {b^{2} e^{2}-11 b c d e +11 d^{2} c^{2}}{5 e^{4} d}-\frac {b \left (b^{2} e^{2}-11 b c d e +11 d^{2} c^{2}\right )}{5 e^{3} d \left (b e -c d \right )}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}+\frac {2 \left (\frac {c^{2}}{e^{3}}-\frac {c \left (b^{2} e^{2}-11 b c d e +11 d^{2} c^{2}\right )}{5 e^{3} d \left (b e -c d \right )}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) \EllipticE \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d \EllipticF \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}\right )}{\sqrt {e x +d}\, x \left (c x +b \right )}\) | \(646\) |
default | \(\text {Expression too large to display}\) | \(1885\) |
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 = 0.50, size = 781, normalized size = 2.21 \begin {gather*} -\frac {2 \, {\left ({\left (16 \, c^{3} d^{6} + b^{3} x^{3} e^{6} + 3 \, {\left (2 \, b^{2} c d x^{3} + b^{3} d x^{2}\right )} e^{5} - 3 \, {\left (8 \, b c^{2} d^{2} x^{3} - 6 \, b^{2} c d^{2} x^{2} - b^{3} d^{2} x\right )} e^{4} + {\left (16 \, c^{3} d^{3} x^{3} - 72 \, b c^{2} d^{3} x^{2} + 18 \, b^{2} c d^{3} x + b^{3} d^{3}\right )} e^{3} + 6 \, {\left (8 \, c^{3} d^{4} x^{2} - 12 \, b c^{2} d^{4} x + b^{2} c d^{4}\right )} e^{2} + 24 \, {\left (2 \, c^{3} d^{5} x - b c^{2} d^{5}\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 (16 \, c^{3} d^{5} e + b^{2} c x^{3} e^{6} - {\left (16 \, b c^{2} d x^{3} - 3 \, b^{2} c d x^{2}\right )} e^{5} + {\left (16 \, c^{3} d^{2} x^{3} - 48 \, b c^{2} d^{2} x^{2} + 3 \, b^{2} c d^{2} x\right )} e^{4} + {\left (48 \, c^{3} d^{3} x^{2} - 48 \, b c^{2} d^{3} x + b^{2} c d^{3}\right )} e^{3} + 16 \, {\left (3 \, c^{3} d^{4} x - b c^{2} d^{4}\right )} e^{2}\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 \, {\left (8 \, c^{3} d^{4} e^{2} - 11 \, b c^{2} d x^{2} e^{5} + b^{2} c x^{2} e^{6} + {\left (11 \, c^{3} d^{2} x^{2} - 16 \, b c^{2} d^{2} x\right )} e^{4} + {\left (18 \, c^{3} d^{3} x - 7 \, b c^{2} d^{3}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )}}{15 \, {\left (c^{2} d^{5} e^{5} - b c d x^{3} e^{9} + {\left (c^{2} d^{2} x^{3} - 3 \, b c d^{2} x^{2}\right )} e^{8} + 3 \, {\left (c^{2} d^{3} x^{2} - b c d^{3} x\right )} e^{7} + {\left (3 \, c^{2} d^{4} x - b c d^{4}\right )} e^{6}\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 {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \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 (c\,x^2+b\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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