Optimal. Leaf size=567 \[ \frac {8 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^2 \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)^2}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (c d-b e)}+\frac {2 \left (4 c x \left (b^3 e^3-6 b c^2 d^2 e+4 c^3 d^3\right )+b (c d-b e) \left (-4 b^2 e^2-3 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)^2}+\frac {2 e \sqrt {b x+c x^2} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right )}{3 b^4 d^3 \sqrt {d+e x} (c d-b e)^3}-\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^3 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^3} \]
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Rubi [A] time = 0.65, antiderivative size = 567, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {740, 822, 834, 843, 715, 112, 110, 117, 116} \[ \frac {2 e \sqrt {b x+c x^2} \left (9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4-32 b c^3 d^3 e+16 c^4 d^4\right )}{3 b^4 d^3 \sqrt {d+e x} (c d-b e)^3}+\frac {2 \left (4 c x \left (b^3 e^3-6 b c^2 d^2 e+4 c^3 d^3\right )+b (c d-b e) \left (-4 b^2 e^2-3 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)^2}+\frac {8 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^2 \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)^2}-\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^3 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^3}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (c d-b e)} \]
Antiderivative was successfully verified.
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Rule 110
Rule 112
Rule 116
Rule 117
Rule 715
Rule 740
Rule 822
Rule 834
Rule 843
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (8 c^2 d^2-3 b c d e-4 b^2 e^2\right )+\frac {5}{2} c e (2 c d-b e) x}{(d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2 d (c d-b e)}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b (c d-b e) \left (8 c^2 d^2-3 b c d e-4 b^2 e^2\right )+4 c \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^2}}+\frac {4 \int \frac {\frac {1}{4} b e \left (8 c^3 d^3-9 b c^2 d^2 e-3 b^2 c d e^2+8 b^3 e^3\right )+c e \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx}{3 b^4 d^2 (c d-b e)^2}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b (c d-b e) \left (8 c^2 d^2-3 b c d e-4 b^2 e^2\right )+4 c \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^2}}+\frac {2 e \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {8 \int \frac {\frac {1}{8} b c d e \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3\right )+\frac {1}{8} c e \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 d^3 (c d-b e)^3}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b (c d-b e) \left (8 c^2 d^2-3 b c d e-4 b^2 e^2\right )+4 c \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^2}}+\frac {2 e \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 \sqrt {d+e x}}+\frac {\left (4 c (2 c d-b e) \left (2 c^2 d^2-2 b c d e-b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 d^2 (c d-b e)^2}-\frac {\left (c \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 b^4 d^3 (c d-b e)^3}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b (c d-b e) \left (8 c^2 d^2-3 b c d e-4 b^2 e^2\right )+4 c \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^2}}+\frac {2 e \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 \sqrt {d+e x}}+\frac {\left (4 c (2 c d-b e) \left (2 c^2 d^2-2 b c d e-b^2 e^2\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 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}-\frac {\left (c \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{3 b^4 d^3 (c d-b e)^3 \sqrt {b x+c x^2}}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b (c d-b e) \left (8 c^2 d^2-3 b c d e-4 b^2 e^2\right )+4 c \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^2}}+\frac {2 e \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {\left (c \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\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 d^3 (c d-b e)^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (4 c (2 c d-b e) \left (2 c^2 d^2-2 b c d e-b^2 e^2\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 d^2 (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b (c d-b e) \left (8 c^2 d^2-3 b c d e-4 b^2 e^2\right )+4 c \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^2}}+\frac {2 e \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {2 \sqrt {c} \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\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} d^3 (c d-b e)^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {8 \sqrt {c} (2 c d-b e) \left (2 c^2 d^2-2 b c d e-b^2 e^2\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} d^2 (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 1.66, size = 504, normalized size = 0.89 \[ -\frac {2 \left (b \left (3 b^4 e^5 x^2 (b+c x)^2+b c^4 d^3 x^2 (d+e x) (b e-c d)-c^4 d^3 x^2 (b+c x) (d+e x) (8 c d-13 b e)+b d (b+c x)^2 (d+e x) (c d-b e)^3-x (b+c x)^2 (d+e x) (c d-b e)^3 (5 b e+8 c d)\right )+c x \sqrt {\frac {b}{c}} (b+c x) \left (-i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (-8 b^4 e^4+11 b^3 c d e^3+6 b^2 c^2 d^2 e^2-17 b c^3 d^3 e+8 c^4 d^4\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+\sqrt {\frac {b}{c}} (b+c x) (d+e x) \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right )\right )\right )}{3 b^5 d^3 (x (b+c x))^{3/2} \sqrt {d+e x} (c d-b e)^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x} \sqrt {e x + d}}{c^{3} e^{2} x^{8} + b^{3} d^{2} x^{3} + {\left (2 \, c^{3} d e + 3 \, b c^{2} e^{2}\right )} x^{7} + {\left (c^{3} d^{2} + 6 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} x^{6} + {\left (3 \, b c^{2} d^{2} + 6 \, b^{2} c d e + b^{3} e^{2}\right )} x^{5} + {\left (3 \, b^{2} c d^{2} + 2 \, b^{3} d e\right )} x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 2189, normalized size = 3.86 \[ \text {result too large to display} \]
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 \[ \int \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (c\,x^2+b\,x\right )}^{5/2}\,{\left (d+e\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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