Optimal. Leaf size=540 \[ -\frac {4 b e \left (1-c^2 x^2\right )}{15 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x (d+e x)^{3/2}}-\frac {16 b c e \left (1-c^2 x^2\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b e \left (1-c^2 x^2\right )}{5 c d^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {4 b \left (7 c^2 d^2-3 e^2\right ) \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 \left (c^2 d^3-d e^2\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 c d^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
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Rubi [A]
time = 0.53, antiderivative size = 637, normalized size of antiderivative = 1.18, number of steps
used = 19, number of rules used = 14, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {5334, 1588,
972, 759, 849, 858, 733, 435, 430, 21, 947, 174, 552, 551} \begin {gather*} -\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {4 b \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 d x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}+\frac {16 b c^2 \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right )^2 \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 d^2 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 c d^2 e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {16 b c e \left (1-c^2 x^2\right )}{15 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right )^2 \sqrt {d+e x}}-\frac {4 b e \left (1-c^2 x^2\right )}{5 c d^2 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {4 b e \left (1-c^2 x^2\right )}{15 c d x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 174
Rule 430
Rule 435
Rule 551
Rule 552
Rule 733
Rule 759
Rule 849
Rule 858
Rule 947
Rule 972
Rule 1588
Rule 5334
Rubi steps
\begin {align*} \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^{7/2}} \, dx &=-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {(2 b) \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^2 (d+e x)^{5/2}} \, dx}{5 c e}\\ &=-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{x (d+e x)^{5/2} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \left (-\frac {e}{d (d+e x)^{5/2} \sqrt {-\frac {1}{c^2}+x^2}}-\frac {e}{d^2 (d+e x)^{3/2} \sqrt {-\frac {1}{c^2}+x^2}}+\frac {1}{d^2 x \sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}}\right ) \, dx}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c d^2 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{(d+e x)^{5/2} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c d \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c d^2 e \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {4 b e \left (1-c^2 x^2\right )}{15 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x (d+e x)^{3/2}}-\frac {4 b e \left (1-c^2 x^2\right )}{5 c d^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {\left (4 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {-\frac {d}{2}-\frac {e x}{2}}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c d^2 \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (4 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {-\frac {3 d}{2}+\frac {e x}{2}}{(d+e x)^{3/2} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{15 c d \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \sqrt {d+e x}} \, dx}{5 c d^2 e \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {4 b e \left (1-c^2 x^2\right )}{15 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x (d+e x)^{3/2}}-\frac {16 b c e \left (1-c^2 x^2\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b e \left (1-c^2 x^2\right )}{5 c d^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {\left (8 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {\frac {1}{4} \left (3 d^2+\frac {e^2}{c^2}\right )+d e x}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{15 c d \left (d^2-\frac {e^2}{c^2}\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c d^2 \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {d+\frac {e}{c}-\frac {e x^2}{c}}} \, dx,x,\sqrt {1-c x}\right )}{5 c d^2 e \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {4 b e \left (1-c^2 x^2\right )}{15 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x (d+e x)^{3/2}}-\frac {16 b c e \left (1-c^2 x^2\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b e \left (1-c^2 x^2\right )}{5 c d^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {\left (8 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {-\frac {1}{c^2}+x^2}} \, dx}{15 c \left (d^2-\frac {e^2}{c^2}\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (2 b \left (-d^2+\frac {e^2}{c^2}\right ) \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{15 c d \left (d^2-\frac {e^2}{c^2}\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {1-\frac {e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\sqrt {1-c x}\right )}{5 c d^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (4 b \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {2 e x^2}{c \left (d+\frac {e}{c}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{5 c^2 d^2 \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {d+e x}{d+\frac {e}{c}}}}\\ &=-\frac {4 b e \left (1-c^2 x^2\right )}{15 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x (d+e x)^{3/2}}-\frac {16 b c e \left (1-c^2 x^2\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b e \left (1-c^2 x^2\right )}{5 c d^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {4 b \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 c d^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (16 b \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {2 e x^2}{c \left (d+\frac {e}{c}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{15 c^2 \left (d^2-\frac {e^2}{c^2}\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {d+e x}{d+\frac {e}{c}}}}+\frac {\left (4 b \left (-d^2+\frac {e^2}{c^2}\right ) \sqrt {\frac {d+e x}{d+\frac {e}{c}}} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1-\frac {2 e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{15 c^2 d \left (d^2-\frac {e^2}{c^2}\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=-\frac {4 b e \left (1-c^2 x^2\right )}{15 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x (d+e x)^{3/2}}-\frac {16 b c e \left (1-c^2 x^2\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b e \left (1-c^2 x^2\right )}{5 c d^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {16 b c^2 \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 c d^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 28.90, size = 407, normalized size = 0.75 \begin {gather*} \frac {2 \left (-\frac {3 a}{(d+e x)^{5/2}}+\frac {2 b c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \left (-e^2 (4 d+3 e x)+c^2 d^2 (8 d+7 e x)\right )}{\left (c^2 d^3-d e^2\right )^2 (d+e x)^{3/2}}-\frac {3 b \sec ^{-1}(c x)}{(d+e x)^{5/2}}+\frac {2 i b \sqrt {\frac {e (1+c x)}{-c d+e}} \sqrt {\frac {e-c e x}{c d+e}} \left (c d \left (7 c^2 d^2-3 e^2\right ) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right )|\frac {c d+e}{c d-e}\right )-c d \left (6 c^2 d^2-c d e-3 e^2\right ) F\left (i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right )|\frac {c d+e}{c d-e}\right )-3 (c d-e) (c d+e)^2 \Pi \left (1+\frac {e}{c d};i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right )|\frac {c d+e}{c d-e}\right )\right )}{d^3 (c d-e) \left (-\frac {c}{c d+e}\right )^{3/2} (c d+e)^3 \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{15 e} \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. \(1619\) vs.
\(2(491)=982\).
time = 0.40, size = 1620, normalized size = 3.00
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1620\) |
default | \(\text {Expression too large to display}\) | \(1620\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{{\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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