Optimal. Leaf size=540 \[ -\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right )}{15 d x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}-\frac{2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac{4 b e \left (1-c^2 x^2\right )}{5 c d^2 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}-\frac{16 b c e \left (1-c^2 x^2\right )}{15 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right )^2 \sqrt{d+e x}}-\frac{4 b e \left (1-c^2 x^2\right )}{15 c d x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}+\frac{4 b \sqrt{1-c^2 x^2} \left (7 c^2 d^2-3 e^2\right ) \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^3-d e^2\right )^2 \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{5 c d^2 e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.710878, 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 = {5226, 1574, 958, 745, 835, 844, 719, 424, 419, 21, 933, 168, 538, 537} \[ -\frac{2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac{16 b c e \left (1-c^2 x^2\right )}{15 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right )^2 \sqrt{d+e x}}-\frac{4 b e \left (1-c^2 x^2\right )}{5 c d^2 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}-\frac{4 b e \left (1-c^2 x^2\right )}{15 c d x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 d x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}+\frac{16 b c^2 \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right )^2 \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{4 b \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{5 d^2 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{5 c d^2 e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5226
Rule 1574
Rule 958
Rule 745
Rule 835
Rule 844
Rule 719
Rule 424
Rule 419
Rule 21
Rule 933
Rule 168
Rule 538
Rule 537
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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [C] time = 8.30249, size = 407, normalized size = 0.75 \[ \frac{2 \left (\frac{2 i b \sqrt{\frac{e (c x+1)}{e-c d}} \sqrt{\frac{e-c e x}{c d+e}} \left (-c d \left (6 c^2 d^2-c d e-3 e^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right ),\frac{c d+e}{c d-e}\right )+c d \left (7 c^2 d^2-3 e^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right )|\frac{c d+e}{c d-e}\right )-3 (c d-e) (c d+e)^2 \Pi \left (\frac{e}{c d}+1;i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right )|\frac{c d+e}{c d-e}\right )\right )}{d^3 x \sqrt{1-\frac{1}{c^2 x^2}} (c d-e) \left (-\frac{c}{c d+e}\right )^{3/2} (c d+e)^3}-\frac{3 a}{(d+e x)^{5/2}}+\frac{2 b c e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2 (8 d+7 e x)-e^2 (4 d+3 e x)\right )}{\left (c^2 d^3-d e^2\right )^2 (d+e x)^{3/2}}-\frac{3 b \sec ^{-1}(c x)}{(d+e x)^{5/2}}\right )}{15 e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.276, size = 1640, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arcsec}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]