Optimal. Leaf size=350 \[ -\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^5 (2 c d-b e)}+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (3 b e g-10 c d g+4 c e f)}{3 e^2 (d+e x)^3 (2 c d-b e)}+\frac {5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (3 b e g-10 c d g+4 c e f)}{6 e^2 (d+e x) (2 c d-b e)}+\frac {5 c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (3 b e g-10 c d g+4 c e f)}{4 e^2}+\frac {5 \sqrt {c} (2 c d-b e) (3 b e g-10 c d g+4 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 e^2} \]
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Rubi [A] time = 0.63, antiderivative size = 350, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {792, 662, 664, 621, 204} \[ -\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^5 (2 c d-b e)}+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (3 b e g-10 c d g+4 c e f)}{3 e^2 (d+e x)^3 (2 c d-b e)}+\frac {5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (3 b e g-10 c d g+4 c e f)}{6 e^2 (d+e x) (2 c d-b e)}+\frac {5 c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (3 b e g-10 c d g+4 c e f)}{4 e^2}+\frac {5 \sqrt {c} (2 c d-b e) (3 b e g-10 c d g+4 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 e^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 621
Rule 662
Rule 664
Rule 792
Rubi steps
\begin {align*} \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx &=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}-\frac {(4 c e f-10 c d g+3 b e g) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx}{3 e (2 c d-b e)}\\ &=\frac {2 (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac {(5 c (4 c e f-10 c d g+3 b e g)) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx}{3 e (2 c d-b e)}\\ &=\frac {5 c (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 e^2 (2 c d-b e) (d+e x)}+\frac {2 (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac {(5 c (4 c e f-10 c d g+3 b e g)) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{d+e x} \, dx}{4 e}\\ &=\frac {5 c (4 c e f-10 c d g+3 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2}+\frac {5 c (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 e^2 (2 c d-b e) (d+e x)}+\frac {2 (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac {(5 c (2 c d-b e) (4 c e f-10 c d g+3 b e g)) \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{8 e}\\ &=\frac {5 c (4 c e f-10 c d g+3 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2}+\frac {5 c (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 e^2 (2 c d-b e) (d+e x)}+\frac {2 (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac {(5 c (2 c d-b e) (4 c e f-10 c d g+3 b e g)) \operatorname {Subst}\left (\int \frac {1}{-4 c e^2-x^2} \, dx,x,\frac {-b e^2-2 c e^2 x}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}\right )}{4 e}\\ &=\frac {5 c (4 c e f-10 c d g+3 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2}+\frac {5 c (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 e^2 (2 c d-b e) (d+e x)}+\frac {2 (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac {5 \sqrt {c} (2 c d-b e) (4 c e f-10 c d g+3 b e g) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 e^2}\\ \end {align*}
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Mathematica [C] time = 0.20, size = 173, normalized size = 0.49 \[ \frac {2 ((d+e x) (c (d-e x)-b e))^{5/2} \left (\frac {(d+e x) (b e-2 c d)^2 (3 b e g-10 c d g+4 c e f) \, _2F_1\left (-\frac {5}{2},-\frac {1}{2};\frac {1}{2};\frac {c (d+e x)}{2 c d-b e}\right )}{\sqrt {\frac {b e-c d+c e x}{b e-2 c d}}}+(e f-d g) (b e-c d+c e x)^3\right )}{3 e^2 (d+e x)^4 (2 c d-b e) (b e-c d+c e x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 7.44, size = 951, normalized size = 2.72 \[ \left [-\frac {15 \, {\left ({\left (4 \, {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} f - {\left (20 \, c^{2} d^{2} e^{2} - 16 \, b c d e^{3} + 3 \, b^{2} e^{4}\right )} g\right )} x^{2} + 4 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} f - {\left (20 \, c^{2} d^{4} - 16 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2}\right )} g + 2 \, {\left (4 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} f - {\left (20 \, c^{2} d^{3} e - 16 \, b c d^{2} e^{2} + 3 \, b^{2} d e^{3}\right )} g\right )} x\right )} \sqrt {-c} \log \left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2} - 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) - 4 \, {\left (6 \, c^{2} e^{3} g x^{3} + 3 \, {\left (4 \, c^{2} e^{3} f - {\left (16 \, c^{2} d e^{2} - 9 \, b c e^{3}\right )} g\right )} x^{2} + 4 \, {\left (23 \, c^{2} d^{2} e - 6 \, b c d e^{2} - 2 \, b^{2} e^{3}\right )} f - {\left (236 \, c^{2} d^{3} - 147 \, b c d^{2} e + 16 \, b^{2} d e^{2}\right )} g + 2 \, {\left (4 \, {\left (17 \, c^{2} d e^{2} - 7 \, b c e^{3}\right )} f - {\left (161 \, c^{2} d^{2} e - 103 \, b c d e^{2} + 12 \, b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{48 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}, -\frac {15 \, {\left ({\left (4 \, {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} f - {\left (20 \, c^{2} d^{2} e^{2} - 16 \, b c d e^{3} + 3 \, b^{2} e^{4}\right )} g\right )} x^{2} + 4 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} f - {\left (20 \, c^{2} d^{4} - 16 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2}\right )} g + 2 \, {\left (4 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} f - {\left (20 \, c^{2} d^{3} e - 16 \, b c d^{2} e^{2} + 3 \, b^{2} d e^{3}\right )} g\right )} x\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) - 2 \, {\left (6 \, c^{2} e^{3} g x^{3} + 3 \, {\left (4 \, c^{2} e^{3} f - {\left (16 \, c^{2} d e^{2} - 9 \, b c e^{3}\right )} g\right )} x^{2} + 4 \, {\left (23 \, c^{2} d^{2} e - 6 \, b c d e^{2} - 2 \, b^{2} e^{3}\right )} f - {\left (236 \, c^{2} d^{3} - 147 \, b c d^{2} e + 16 \, b^{2} d e^{2}\right )} g + 2 \, {\left (4 \, {\left (17 \, c^{2} d e^{2} - 7 \, b c e^{3}\right )} f - {\left (161 \, c^{2} d^{2} e - 103 \, b c d e^{2} + 12 \, b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{24 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 5227, normalized size = 14.93 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^5} \,d x \]
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 \[ \int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (f + g x\right )}{\left (d + e x\right )^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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