Optimal. Leaf size=310 \[ -\frac {2 (d+e x)^{3/2} (b d+(2 c d-b e) x)}{b^2 \sqrt {b x+c x^2}}+\frac {2 e (2 c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}{b^2 c}+\frac {4 \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} c^{3/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 d (c d-b e) (2 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} c^{3/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \]
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Rubi [A]
time = 0.23, antiderivative size = 310, 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 = {752, 846, 857,
729, 113, 111, 118, 117} \begin {gather*} \frac {4 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} c^{3/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) F\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} c^{3/2} \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 (d+e x)^{3/2} (x (2 c d-b e)+b d)}{b^2 \sqrt {b x+c x^2}}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x} (2 c d-b e)}{b^2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 111
Rule 113
Rule 117
Rule 118
Rule 729
Rule 752
Rule 846
Rule 857
Rubi steps
\begin {align*} \int \frac {(d+e x)^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (d+e x)^{3/2} (b d+(2 c d-b e) x)}{b^2 \sqrt {b x+c x^2}}-\frac {2 \int \frac {\sqrt {d+e x} \left (-\frac {3}{2} b d e-\frac {3}{2} e (2 c d-b e) x\right )}{\sqrt {b x+c x^2}} \, dx}{b^2}\\ &=-\frac {2 (d+e x)^{3/2} (b d+(2 c d-b e) x)}{b^2 \sqrt {b x+c x^2}}+\frac {2 e (2 c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}{b^2 c}-\frac {4 \int \frac {-\frac {3}{4} b d e (c d+b e)-\frac {3}{2} e \left (c^2 d^2-b c d e+b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^2 c}\\ &=-\frac {2 (d+e x)^{3/2} (b d+(2 c d-b e) x)}{b^2 \sqrt {b x+c x^2}}+\frac {2 e (2 c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}{b^2 c}-\frac {(d (c d-b e) (2 c d-b e)) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{b^2 c}+\frac {\left (2 \left (c^2 d^2-b c d e+b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{b^2 c}\\ &=-\frac {2 (d+e x)^{3/2} (b d+(2 c d-b e) x)}{b^2 \sqrt {b x+c x^2}}+\frac {2 e (2 c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}{b^2 c}-\frac {\left (d (c d-b e) (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}{b^2 c \sqrt {b x+c x^2}}+\frac {\left (2 \left (c^2 d^2-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}{b^2 c \sqrt {b x+c x^2}}\\ &=-\frac {2 (d+e x)^{3/2} (b d+(2 c d-b e) x)}{b^2 \sqrt {b x+c x^2}}+\frac {2 e (2 c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}{b^2 c}+\frac {\left (2 \left (c^2 d^2-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}{b^2 c \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left (d (c d-b e) (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}{b^2 c \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=-\frac {2 (d+e x)^{3/2} (b d+(2 c d-b e) x)}{b^2 \sqrt {b x+c x^2}}+\frac {2 e (2 c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}{b^2 c}+\frac {4 \left (c^2 d^2-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 )}{(-b)^{3/2} c^{3/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 d (c d-b e) (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 )}{(-b)^{3/2} 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 = 9.61, size = 262, normalized size = 0.85 \begin {gather*} \frac {2 b (d+e x) \left (c^2 d^2+2 b^2 e^2+b c e (-2 d+e x)\right )+4 i \sqrt {\frac {b}{c}} c e \left (c^2 d^2-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 )-2 i \sqrt {\frac {b}{c}} c e \left (c^2 d^2-3 b c d e+2 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 )}{b^2 c^2 \sqrt {x (b+c x)} \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. \(651\) vs.
\(2(264)=528\).
time = 0.46, size = 652, normalized size = 2.10
method | result | size |
elliptic | \(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (-\frac {2 \left (c e \,x^{2}+c d x \right ) \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}{b^{2} c^{2} \sqrt {\left (\frac {b}{c}+x \right ) \left (c e \,x^{2}+c d x \right )}}-\frac {2 \left (c e \,x^{2}+b e x +c d x +b d \right ) d^{2}}{b^{2} \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}+\frac {2 \left (-\frac {e^{2} \left (b e -3 c d \right )}{c^{2}}+\frac {\left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right ) \left (b e -c d \right )}{c^{2} b^{2}}+\frac {d \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}{c \,b^{2}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}+\frac {2 \left (\frac {e^{3}}{c}+\frac {\left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right ) e}{c \,b^{2}}+\frac {c \,d^{2} e}{b^{2}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) \EllipticE \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d \EllipticF \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\) | \(557\) |
default | \(-\frac {2 \left (\sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} c d \,e^{2}-3 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c^{2} d^{2} e +2 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{3} d^{3}+2 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{4} e^{3}-4 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} c d \,e^{2}+4 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c^{2} d^{2} e -2 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{3} d^{3}+b^{2} c^{2} e^{3} x^{2}-2 b \,c^{3} d \,e^{2} x^{2}+2 c^{4} d^{2} e \,x^{2}+b^{2} c^{2} d \,e^{2} x -b \,c^{3} d^{2} e x +2 c^{4} d^{3} x +c^{3} b \,d^{3}\right ) \sqrt {x \left (c x +b \right )}}{x \left (c x +b \right ) c^{3} b^{2} \sqrt {e x +d}}\) | \(652\) |
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 = 0.72, size = 517, normalized size = 1.67 \begin {gather*} -\frac {2 \, {\left ({\left (2 \, c^{4} d^{3} x^{2} + 2 \, b c^{3} d^{3} x + 2 \, {\left (b^{3} c x^{2} + b^{4} x\right )} e^{3} - 3 \, {\left (b^{2} c^{2} d x^{2} + b^{3} c d x\right )} e^{2} - 3 \, {\left (b c^{3} d^{2} x^{2} + b^{2} c^{2} d^{2} x\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 ) + 6 \, {\left ({\left (b^{2} c^{2} x^{2} + b^{3} c x\right )} e^{3} - {\left (b c^{3} d x^{2} + b^{2} c^{2} d x\right )} e^{2} + {\left (c^{4} d^{2} x^{2} + b c^{3} d^{2} x\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 \, {\left (2 \, b c^{3} d x e^{2} - b^{2} c^{2} x e^{3} - {\left (2 \, c^{4} d^{2} x + b c^{3} d^{2}\right )} e\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{3 \, {\left (b^{2} c^{4} x^{2} + b^{3} c^{3} x\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 (d + e x\right )^{\frac {5}{2}}}{\left (x \left (b + c x\right )\right )^{\frac {3}{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 (d+e\,x\right )}^{5/2}}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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