Optimal. Leaf size=426 \[ \frac{\sqrt{\frac{c x^2}{a}+1} \left (3 c d^2-5 a e^2\right ) \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 \sqrt{-a} c^{5/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{e \sqrt{a+c x^2} \sqrt{d+e x} \left (3 c d^2-5 a e^2\right )}{3 a c^2}-\frac{d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (3 c d^2-29 a e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 \sqrt{-a} c^{3/2} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{(d+e x)^{5/2} (a e-c d x)}{a c \sqrt{a+c x^2}}-\frac{d e \sqrt{a+c x^2} (d+e x)^{3/2}}{a c} \]
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Rubi [A] time = 0.469331, antiderivative size = 426, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {739, 833, 844, 719, 424, 419} \[ -\frac{e \sqrt{a+c x^2} \sqrt{d+e x} \left (3 c d^2-5 a e^2\right )}{3 a c^2}+\frac{\sqrt{\frac{c x^2}{a}+1} \left (3 c d^2-5 a e^2\right ) \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 \sqrt{-a} c^{5/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (3 c d^2-29 a e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 \sqrt{-a} c^{3/2} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{(d+e x)^{5/2} (a e-c d x)}{a c \sqrt{a+c x^2}}-\frac{d e \sqrt{a+c x^2} (d+e x)^{3/2}}{a c} \]
Antiderivative was successfully verified.
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Rule 739
Rule 833
Rule 844
Rule 719
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{(d+e x)^{7/2}}{\left (a+c x^2\right )^{3/2}} \, dx &=-\frac{(a e-c d x) (d+e x)^{5/2}}{a c \sqrt{a+c x^2}}+\frac{\int \frac{(d+e x)^{3/2} \left (\frac{5 a e^2}{2}-\frac{5}{2} c d e x\right )}{\sqrt{a+c x^2}} \, dx}{a c}\\ &=-\frac{(a e-c d x) (d+e x)^{5/2}}{a c \sqrt{a+c x^2}}-\frac{d e (d+e x)^{3/2} \sqrt{a+c x^2}}{a c}+\frac{2 \int \frac{\sqrt{d+e x} \left (10 a c d e^2-\frac{5}{4} c e \left (3 c d^2-5 a e^2\right ) x\right )}{\sqrt{a+c x^2}} \, dx}{5 a c^2}\\ &=-\frac{(a e-c d x) (d+e x)^{5/2}}{a c \sqrt{a+c x^2}}-\frac{e \left (3 c d^2-5 a e^2\right ) \sqrt{d+e x} \sqrt{a+c x^2}}{3 a c^2}-\frac{d e (d+e x)^{3/2} \sqrt{a+c x^2}}{a c}+\frac{4 \int \frac{\frac{5}{8} a c e^2 \left (27 c d^2-5 a e^2\right )-\frac{5}{8} c^2 d e \left (3 c d^2-29 a e^2\right ) x}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{15 a c^3}\\ &=-\frac{(a e-c d x) (d+e x)^{5/2}}{a c \sqrt{a+c x^2}}-\frac{e \left (3 c d^2-5 a e^2\right ) \sqrt{d+e x} \sqrt{a+c x^2}}{3 a c^2}-\frac{d e (d+e x)^{3/2} \sqrt{a+c x^2}}{a c}-\frac{\left (d \left (3 c d^2-29 a e^2\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx}{6 a c}+\frac{\left (\left (3 c d^2-5 a e^2\right ) \left (c d^2+a e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{6 a c^2}\\ &=-\frac{(a e-c d x) (d+e x)^{5/2}}{a c \sqrt{a+c x^2}}-\frac{e \left (3 c d^2-5 a e^2\right ) \sqrt{d+e x} \sqrt{a+c x^2}}{3 a c^2}-\frac{d e (d+e x)^{3/2} \sqrt{a+c x^2}}{a c}-\frac{\left (d \left (3 c d^2-29 a e^2\right ) \sqrt{d+e x} \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 a \sqrt{c} e x^2}{\sqrt{-a} \left (c d-\frac{a \sqrt{c} e}{\sqrt{-a}}\right )}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )}{3 \sqrt{-a} c^{3/2} \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{a+c x^2}}+\frac{\left (\left (3 c d^2-5 a e^2\right ) \left (c d^2+a e^2\right ) \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 a \sqrt{c} e x^2}{\sqrt{-a} \left (c d-\frac{a \sqrt{c} e}{\sqrt{-a}}\right )}}} \, dx,x,\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )}{3 \sqrt{-a} c^{5/2} \sqrt{d+e x} \sqrt{a+c x^2}}\\ &=-\frac{(a e-c d x) (d+e x)^{5/2}}{a c \sqrt{a+c x^2}}-\frac{e \left (3 c d^2-5 a e^2\right ) \sqrt{d+e x} \sqrt{a+c x^2}}{3 a c^2}-\frac{d e (d+e x)^{3/2} \sqrt{a+c x^2}}{a c}-\frac{d \left (3 c d^2-29 a e^2\right ) \sqrt{d+e x} \sqrt{1+\frac{c x^2}{a}} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 \sqrt{-a} c^{3/2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{a+c x^2}}+\frac{\left (3 c d^2-5 a e^2\right ) \left (c d^2+a e^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{1+\frac{c x^2}{a}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 \sqrt{-a} c^{5/2} \sqrt{d+e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 3.47229, size = 586, normalized size = 1.38 \[ \frac{\sqrt{d+e x} \left (\frac{2 \left (\sqrt{a} e (d+e x)^{3/2} \left (-5 i a^{3/2} e^3+27 i \sqrt{a} c d^2 e-29 a \sqrt{c} d e^2+3 c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right ),\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )-d e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (-29 a^2 e^2+a c \left (3 d^2-29 e^2 x^2\right )+3 c^2 d^2 x^2\right )+\sqrt{c} d (d+e x)^{3/2} \left (29 a^{3/2} e^3-3 \sqrt{a} c d^2 e-29 i a \sqrt{c} d e^2+3 i c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{a c^2 e (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}+\frac{10 a e^3}{c^2}+\frac{6 d^3 x}{a}+\frac{2 e \left (-9 d^2-9 d e x+2 e^2 x^2\right )}{c}\right )}{6 \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.35, size = 1362, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{7}{2}}}{{\left (c x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt{c x^{2} + a} \sqrt{e x + d}}{c^{2} x^{4} + 2 \, a c x^{2} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{7}{2}}}{{\left (c x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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